Entering a circle into a graphing calculator can be accomplished by converting the circle to a polar equation and then entering it into the equation editor. Polar equations have an r variable, which is the radial coordinate and a dependent variable, and a θ, which is the angular coordinate and an independent variable. In standard form, a circle equation is written in the form (x - h)^2 + (y - k)^2 = r^2 where (h,k) is the center and r is the radius. An important circle is the unit circle which has the equation x^2 + y^2 = 1 and a radius of 1.
Instructions
1. Convert the expression to polar coordinates and simplify. To convert to polar coordinates, use the conversions x = r*cosθ and x = r*sinθ. For example, the unit circle would become (r*cosθ)^2 + (r*sinθ)^2 = 1. (r*cosθ)^2 + (r*sinθ)^2 is the same as r^2*cos^2(θ) + r^2*sin^2(θ). Pulling out the r^2 factor from both terms on the left-hand side of the equation gives r^2(cos^2(θ) + sin^2(θ)) = 1. By a Pythagorean identity in trigonometry, cos^2(θ) + sin^2(θ) = 1, so the equation becomes r^2 = 1.
2. Solve for the "r" variable. In the unit circle example, take the square root of both sides to get r = √(1).
3. Switch to polar mode on the calculator. On the TI-89, for example, you can do this by pressing the "Mode" button and then pressing arrow "->" key while on the "Graph" menu line. Press "3" to select "Polar." Press "Enter" to save the changes.
4. Enter the equation into the calculator's equation editor. For example, on a TI-89, press the green diamond button and then "F1." Type √(-x^2 + 1) after the "r1=."
5. View the graph. To do this on a TI-89, press the "Graph" button.
Friday, August 9, 2013
Write an Absolute Value on a Graphing Calculator
The absolute value of an integer is difference between the number and zero. For instance, both 8 and minus- 8 have absolute values of 8 because they are both eight spaces away from zero, albeit in opposite directions. In algebra, you may be required to solve equations with absolute values by graphing, or you might need to quickly calculate the absolute value of a numerical expression, such as -5(3)^31. When you are allowed to use a graphing calculator, it is important to understand how to write absolute value on it.
Instructions:
Finding the Absolute Value of a Number or Numerical Expression
1. Press the "ON" button in the lower left-hand corner of the calculator. Then, press the "MATH" button toward the upper left.
2. Use the right arrow to scroll sideways to the "NUM" selection. Then, either press "ENTER" in the lower right-hand corner of the keypad or the number 1 to select the absolute value function.
3. Type in the numerical expression. Then, insert a left-facing parenthesis, located in the middle of the keyboard, to close the absolute value. Hit "ENTER" to get the answer.
Graphing an Absolute Value
1. Press the "ON" button in the lower left-hand corner of the calculator. Then, press the "Y=" in the upper left.
2. Select the "MATH" button and then scroll to the right using the directional arrow to select "NUM."
3. Press "ENTER" or 1. Type in the algebraic equation, such as X - 3. Use the variable button "X,T..." for any variables.
4. Close the equation with a left-facing parenthesis. Then, hit "GRAPH" in the upper right to display the graph of the equation.
Tips & Warnings
- Insert the absolute value before the number or expression that goes in it, not after.
- Type any coefficients or parts of the equation outside of the absolute value before you insert it. For instance, with 3abs(2X - 4), you type the 3, then perform the steps above.
Instructions:
Finding the Absolute Value of a Number or Numerical Expression
1. Press the "ON" button in the lower left-hand corner of the calculator. Then, press the "MATH" button toward the upper left.
2. Use the right arrow to scroll sideways to the "NUM" selection. Then, either press "ENTER" in the lower right-hand corner of the keypad or the number 1 to select the absolute value function.
3. Type in the numerical expression. Then, insert a left-facing parenthesis, located in the middle of the keyboard, to close the absolute value. Hit "ENTER" to get the answer.
Graphing an Absolute Value
1. Press the "ON" button in the lower left-hand corner of the calculator. Then, press the "Y=" in the upper left.
2. Select the "MATH" button and then scroll to the right using the directional arrow to select "NUM."
3. Press "ENTER" or 1. Type in the algebraic equation, such as X - 3. Use the variable button "X,T..." for any variables.
4. Close the equation with a left-facing parenthesis. Then, hit "GRAPH" in the upper right to display the graph of the equation.
Tips & Warnings
- Insert the absolute value before the number or expression that goes in it, not after.
- Type any coefficients or parts of the equation outside of the absolute value before you insert it. For instance, with 3abs(2X - 4), you type the 3, then perform the steps above.
Find an Inverse Matrix on a Graphing Calculator
An inverse matrix is the reciprocal of a matrix. A matrix times its inverse equals 1, which is called the Identity Matrix. For a matrix to have an inverse, it must be a square meaning it has to have the same number of rows and columns. Finding the inverse of a matrix by hand is simple, but it's even easier on a graphing calculator such as the TI-83.
Instructions
1. Press the "MATRX" button from the home page on the graphing calculator.
2. Enter a matrix into the calculator if you do not already have one entered. To enter a matrix, press the arrow that points to the right two times when the cursor highlights letter "A." This will move the top cursor to "Edit." Press "Enter" to select the "Edit" option.
3. Enter the matrix's numbers into the matrix on the calculator. Press "Enter" to move the cursor over one column. Press a button that has a dash symbol within parenthesis [( - )] to move down one row. Repeat the steps until the matrix is completed.
4. Press "2nd" then "Mode" to go back to the home screen. Press "MATRX" to enter the matrix screen again.
5. Press "Enter" when the cursor highlights matrix "A" then press "[x^ -1]." Your screen should read "[A]^ -1." Press "Enter" again to display the inverse matrix.
Instructions
1. Press the "MATRX" button from the home page on the graphing calculator.
2. Enter a matrix into the calculator if you do not already have one entered. To enter a matrix, press the arrow that points to the right two times when the cursor highlights letter "A." This will move the top cursor to "Edit." Press "Enter" to select the "Edit" option.
3. Enter the matrix's numbers into the matrix on the calculator. Press "Enter" to move the cursor over one column. Press a button that has a dash symbol within parenthesis [( - )] to move down one row. Repeat the steps until the matrix is completed.
4. Press "2nd" then "Mode" to go back to the home screen. Press "MATRX" to enter the matrix screen again.
5. Press "Enter" when the cursor highlights matrix "A" then press "[x^ -1]." Your screen should read "[A]^ -1." Press "Enter" again to display the inverse matrix.
Graph With TI Calculators
Texas Instruments is a leading manufacturer of a variety of electronics products. Among their most popular products are their TI calculators which students and educators both use, especially when graphing functions. How to graph with TI calculators can seem challenging at first, but once you learn the basics, it is easily mastered and makes the graphing process much easier than the pen and paper method.
Instructions
1. Press the "Y =" button to input one or more functions to be graphed.
2. Press the "Window" button to denote the viewing variables for the x and y axis.
3. Press the "2nd" button and then the "Format" button to set the graph format settings (e.g. denote whether or not you want the coordinates of the axis displayed, whether you want the grid displayed, etc.).
4. Press the "Graph" button to view your graph.
Tips & Warnings
- Your function must be in slope-intercept form (y = mx + b) in order for your linear equation to graph on your calculator correctly.
- Press the "2nd" button and then the "Graph" button to see a table of the x and y values for each point on your graph.
- Press the up, down, left and right arrow keys to move the cursor on your graph and locate a specific point.
- If your line is not displayed on your graph, press "Window" and adjust your x and y axis settings to larger numbers.
Instructions
1. Press the "Y =" button to input one or more functions to be graphed.
2. Press the "Window" button to denote the viewing variables for the x and y axis.
3. Press the "2nd" button and then the "Format" button to set the graph format settings (e.g. denote whether or not you want the coordinates of the axis displayed, whether you want the grid displayed, etc.).
4. Press the "Graph" button to view your graph.
Tips & Warnings
- Your function must be in slope-intercept form (y = mx + b) in order for your linear equation to graph on your calculator correctly.
- Press the "2nd" button and then the "Graph" button to see a table of the x and y values for each point on your graph.
- Press the up, down, left and right arrow keys to move the cursor on your graph and locate a specific point.
- If your line is not displayed on your graph, press "Window" and adjust your x and y axis settings to larger numbers.
Using a Ti84 Graphing Calculator to Find the Zero Function
The zero function, also referred to as "the zeros of a function," is the point on the Cartesian graph where a particular function crosses the X axis. This information is important because a function can cross the X axis up to as many times as the function has degrees. Furthermore, the zeros of a function will be the same values as the answers to the equations found after factoring the function.
Instructions
1. Press the graphing button in the upper left corner of the TI-84. This will take you to the graphing screen.
2. Ensure all other previously entered equations are cleared.
3. Enter the function into the calculator.
4. Press the "Graph" button. This button is located in the upper right of the keypad, directly below the screen.
5. Ensure that your window settings are configured to allow for the entire function to be seen. Only the direction changes of the function are important.
6. Press the "Trace" button.
7. Use the arrow keys below the "Graph" button to move the cursor to every X position where the line crosses the X axis. Every point where Y equals zero is a "zero function." Take note of the X axis locations to verify any factoring that may be needed later. These zeros may also be verified using synthetic division if needed.
Instructions
1. Press the graphing button in the upper left corner of the TI-84. This will take you to the graphing screen.
2. Ensure all other previously entered equations are cleared.
3. Enter the function into the calculator.
4. Press the "Graph" button. This button is located in the upper right of the keypad, directly below the screen.
5. Ensure that your window settings are configured to allow for the entire function to be seen. Only the direction changes of the function are important.
6. Press the "Trace" button.
7. Use the arrow keys below the "Graph" button to move the cursor to every X position where the line crosses the X axis. Every point where Y equals zero is a "zero function." Take note of the X axis locations to verify any factoring that may be needed later. These zeros may also be verified using synthetic division if needed.
Make a Circle in a Graphic Calculator
A graphic calculator plots graphs based on the mathematical formulae entered by the user. Simple equations can result in both simple and complex graphs, while seemingly complex formulae can produce surprisingly simple results.
To draw a specific shape, all you need is to know its formula. In the case of a circle, this is x^2 + y^2 = the square of the radius of the circle to be drawn. Once this is known, it is easy to make a circle on a graphic calculator.
Instructions
1. Solve the equation for x or y, depending on which version of the equation the calculator requires. For a circle, the result is the same either way, so we will solve for x. As x^2 + y^2 = d, x^2 must equal d - y^2. It therefore follows that x equals the square root of the radius squared, minus y^2. Similarly, y equals the square root of the radius squared, minus x^2.
2. Simple equations can form complex graphs.
Enter the formula into the first plotting line on the calculator. The precise method of doing this will differ between models of graphic calculator, so refer to the manual for precise instructions. For most calculators, the equation for a circle of radius 2 will look like this:
Y=sqrt(4-X^2)
Because the radius is squared, we must use 4 rather than 2.
3. Enter the negative of the formula into the second plotting line on the calculator. This is as simple as putting a minus sign in front of the formula entered in Step 2. Press "Enter" or "Trace" (the exact name of the button will vary between models) to have the calculator draw the circle.
Tips & Warnings
- The circle will be centered on zero and extend out as far as the radius you entered, on both the x and y axes. To offset the circle along the x axis, simply change x^2 to (x-1)^2, or whatever distance you want to offset the circle by. Do the same with the y value to move the circle up or down on the graph.
- Some calculators automatically zoom in, and some don't. As a result, it can be difficult to see circles with large or small radii. To prevent this, either use a value for the radius that gives a circle your calculator can plot without trouble, or consult your manual to see how to zoom in or out.
To draw a specific shape, all you need is to know its formula. In the case of a circle, this is x^2 + y^2 = the square of the radius of the circle to be drawn. Once this is known, it is easy to make a circle on a graphic calculator.
Instructions
1. Solve the equation for x or y, depending on which version of the equation the calculator requires. For a circle, the result is the same either way, so we will solve for x. As x^2 + y^2 = d, x^2 must equal d - y^2. It therefore follows that x equals the square root of the radius squared, minus y^2. Similarly, y equals the square root of the radius squared, minus x^2.
2. Simple equations can form complex graphs.
Enter the formula into the first plotting line on the calculator. The precise method of doing this will differ between models of graphic calculator, so refer to the manual for precise instructions. For most calculators, the equation for a circle of radius 2 will look like this:
Y=sqrt(4-X^2)
Because the radius is squared, we must use 4 rather than 2.
3. Enter the negative of the formula into the second plotting line on the calculator. This is as simple as putting a minus sign in front of the formula entered in Step 2. Press "Enter" or "Trace" (the exact name of the button will vary between models) to have the calculator draw the circle.
Tips & Warnings
- The circle will be centered on zero and extend out as far as the radius you entered, on both the x and y axes. To offset the circle along the x axis, simply change x^2 to (x-1)^2, or whatever distance you want to offset the circle by. Do the same with the y value to move the circle up or down on the graph.
- Some calculators automatically zoom in, and some don't. As a result, it can be difficult to see circles with large or small radii. To prevent this, either use a value for the radius that gives a circle your calculator can plot without trouble, or consult your manual to see how to zoom in or out.
Plot Line Segments on Graphing Calculator
In Algebra class, a student becomes accustomed to using a graphing calculator to graph lines, functions and line segments. You need to be able to graph all three of these without your calculator, but if you want to quickly visualize a line segment, or a portion of a line defined specifically between two coordinates, your graphing calculator can instantaneously create such a graph.
Instructions
1. Access the "draw" menu, and select the line command. The calculator will display the "Line" function, along with open brackets.
2. Enter the coordinates of the endpoints of the line segment, separated by commas, in the form "Line(X1, Y1, X2, Y2)". If, for instance, the coordinates if your line segment are "(0,3)" and "(1,2)", you would enter "Line(0,3,1,2)".
3. Press "Enter" and your calculator will plot the segment.
Instructions
1. Access the "draw" menu, and select the line command. The calculator will display the "Line" function, along with open brackets.
2. Enter the coordinates of the endpoints of the line segment, separated by commas, in the form "Line(X1, Y1, X2, Y2)". If, for instance, the coordinates if your line segment are "(0,3)" and "(1,2)", you would enter "Line(0,3,1,2)".
3. Press "Enter" and your calculator will plot the segment.
Change the Base Log on a Graphing Calculator
Most graphing calculators only include keys for base 10 or base "e" logarithms. In order to use other bases, you'll have to take advantage of the change of base formula. Before handheld calculators were the norm, accountants, scientists and engineers used logarithms to multiply and divide large numbers. Now, logarithms are still taught in schools due to their usefulness in science and statistics. For example, scientists use the Richter scale, which is logarithmic, to measure the magnitude of earthquakes.
Instructions
1. Identify the base of your logarithm. The base is written in subscript next to the log. Assume any logarithm without a subscript is logarithm base 10. For example, the base of Log2(35) is 2.
2. Identify the number you are taking the logarithm of. This is written in parenthesis next to the log. In the expression Log2(35), you're taking the logarithm of 35.
3. Press the "Log" key on your graphing calculator. Enter the number you're taking the logarithm of. If you wanted to find the value of Log2(35), you would enter "Log 35."
4. Press the division key.
5. Press the "Log" key and enter the base of the logarithm. To find the value of Log2(35), you would enter "Log 2".
6. Press the "Enter" key. The calculator will output the value.
Tips & Warnings
- You can use either logarithm base 10 or the natural logarithm to do this calculation, both will result in the same answer.
- The natural logarithm uses Euler's number, "e", as a base. "e" is approximately 2.718.
- Mathematicians write "Log" without a superscript to signify the natural logarithm. Others, such as scientists, use that to denote logarithms in base 10.
Instructions
1. Identify the base of your logarithm. The base is written in subscript next to the log. Assume any logarithm without a subscript is logarithm base 10. For example, the base of Log2(35) is 2.
2. Identify the number you are taking the logarithm of. This is written in parenthesis next to the log. In the expression Log2(35), you're taking the logarithm of 35.
3. Press the "Log" key on your graphing calculator. Enter the number you're taking the logarithm of. If you wanted to find the value of Log2(35), you would enter "Log 35."
4. Press the division key.
5. Press the "Log" key and enter the base of the logarithm. To find the value of Log2(35), you would enter "Log 2".
6. Press the "Enter" key. The calculator will output the value.
Tips & Warnings
- You can use either logarithm base 10 or the natural logarithm to do this calculation, both will result in the same answer.
- The natural logarithm uses Euler's number, "e", as a base. "e" is approximately 2.718.
- Mathematicians write "Log" without a superscript to signify the natural logarithm. Others, such as scientists, use that to denote logarithms in base 10.
Use a Graph Calculator to Find the Slope of M
The slope of a line, often referred to as the variable "m" in mathematics, is a measure of how steep a line is. In architecture, slope is used to refer to the steepness of a roof. A roof may rise 1 foot for every foot of the roof's span, for a slope of 1. In architecture, the slope of a roof is often referred to as the rise over the run. Using a graphing calculator to find the slope of a line requires that you use two points on the line to first determine the rise and the run.
Instructions
1. Enter the equation of a line into your online or desktop graphing calculator. Use 2x + 1 for the line equation in this example. Select the graph or plot function on your graphing calculator to plot the line on the screen.
2. Select an arbitrary point on your line. Call that point the origin reference, or R. Assign the origin reference point a x-coordinate value of 0 and a y-coordinate value of 0. Write down the point in symbolic mathematical notation as R(0,0) for origin reference point at x-coordinate, 0, and y-coordinate, 0.
3. Pick an arbitrary point to the right of reference origin point, R. Call this point the terminal point, or T, for short.
4. Write down the horizontal distance (inches, centimeters or other units your graphing calculator is demarcated in) between the reference origin and the terminal point. Call this distance the "run."
5. Write down the vertical distance (inches, centimeters or other units your graphing calculator is demarcated in) between the reference origin and terminal point. Call this distance the "rise." Prefix the rise with a negative sign if the terminal point, P, is below the reference point, R.
6. Divide the rise, from Step 5, by the run, from Step 4, to determine the slope, m. Verify that the slope, m, you calculated is 2, which is the slope of the line equation you entered in the graphing calculator in Step 1.
Instructions
1. Enter the equation of a line into your online or desktop graphing calculator. Use 2x + 1 for the line equation in this example. Select the graph or plot function on your graphing calculator to plot the line on the screen.
2. Select an arbitrary point on your line. Call that point the origin reference, or R. Assign the origin reference point a x-coordinate value of 0 and a y-coordinate value of 0. Write down the point in symbolic mathematical notation as R(0,0) for origin reference point at x-coordinate, 0, and y-coordinate, 0.
3. Pick an arbitrary point to the right of reference origin point, R. Call this point the terminal point, or T, for short.
4. Write down the horizontal distance (inches, centimeters or other units your graphing calculator is demarcated in) between the reference origin and the terminal point. Call this distance the "run."
5. Write down the vertical distance (inches, centimeters or other units your graphing calculator is demarcated in) between the reference origin and terminal point. Call this distance the "rise." Prefix the rise with a negative sign if the terminal point, P, is below the reference point, R.
6. Divide the rise, from Step 5, by the run, from Step 4, to determine the slope, m. Verify that the slope, m, you calculated is 2, which is the slope of the line equation you entered in the graphing calculator in Step 1.
Enter Function Domains on a Graphing Calculator
Many times in mathematics, it is necessary to restrict a function's domain. This is especially important when dealing with trigonometric, absolute-value and radical functions. While doing this on a piece of paper simply requires writing the domain next to the function -- e.g., f(x) = |x|, (0, infinity], performing it on a TI-83 or TI-84 graphing calculator requires different steps. In particular, after you have submitted your equation, you must convert the domain to an inequality and then submit it.
Instructions
1. Write out the domain using inequality signs. For instance, given the domain (0, infinity] for a function Y(x), write x > 0. Given the domain (-5, 5), write 5 > x > -5.
2. Type your equation in the graphing calculator but do not submit it.
3. Add a division sign and the inequality (in quotes) after the equation. Given the equation Y(x) = |x| and the domain (-5, 5), your screen would look like this: Y1 = |x| / (5>x>-5). Likewise, given the equation Y(x) = |x| and the domain (0, infinity], your screen would look like this: Y1 = |x| / (x>0).
Instructions
1. Write out the domain using inequality signs. For instance, given the domain (0, infinity] for a function Y(x), write x > 0. Given the domain (-5, 5), write 5 > x > -5.
2. Type your equation in the graphing calculator but do not submit it.
3. Add a division sign and the inequality (in quotes) after the equation. Given the equation Y(x) = |x| and the domain (-5, 5), your screen would look like this: Y1 = |x| / (5>x>-5). Likewise, given the equation Y(x) = |x| and the domain (0, infinity], your screen would look like this: Y1 = |x| / (x>0).
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