Solving Quadratic Equations – Graphing Method

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Presentation transcript:

Solving Quadratic Equations – Graphing Method This presentation explains how to solve quadratic equations graphically. The first step is to get the quadratic equation into the following form: Note that all non-zero terms have been moved to the left side of the equation, with zero on the right side.

Let y1 equal the left side of the equation: Since the right side of the equation is zero, y1 must equal zero. Recall that anytime the y-value is zero, the ordered pair (x,0) is a point on the x-axis. The solutions to the equation are the x-values for which y1 (left side of equation) is equal to zero, and thus the x-values of the x-intercepts on the graph of y1.

Finding the x-intercepts: Method 1: Trace - for integer answers. Enter y1 into the calculator and graph. Use the Zoom and Window features if needed to get a better estimate of the x-intercepts. Press Trace and enter the estimated x-value. If the corresponding y-value is zero, then you know the given x-value is an x-intercept.

Finding the x-intercepts: Method 2: Intersect - for non-integer answers. Enter y1 into the y= screen. Enter y2=0 Note that the graph of y2 is a horizontal line which is the x-axis. Now use the 2nd Calc|Intersect method on the calculator to find the points of intersection. Note that these points are x-intercepts.

Example 1- integer answers: Use the graphing method to solve the quadratic equation: Simplify, with all terms on the left hand side.

Let y1 equal the left hand side and enter the result into the calculator. Press Zoom-6 to graph Since it appears that the intercepts may be integers, use the Trace method.

Press Trace and then enter -4. Since the y-value is 0, (-4,0) is an x-intercept, and -4 is a solution to the equation. Now repeat the process with x = 3, the intercept on the right. Again, the y-value is 0. The solutions to the equation are x = -4, 3

Example 2- non-integer answers: Use the graphing method to solve the quadratic equation. Round answers to the nearest thousandth: Simplify, with all terms on the left hand side.

Let y1 equal the left hand side and enter the result into the calculator. Press Zoom-6 to graph Try the Trace method to see if the intercepts are integers.

Press Trace and then the x-value on the left, -1. Since the y-value is not 0, (-1,0) is not an x-intercept. Use the intersect method to find approximate x-intercepts. Enter y2 = 0, and graph. Notice that nothing has appeared to change. This is because y2 = 0 is right on the x-axis.

Press 2nd|Calc|Intersect. The calculator asks three questions. Speed up the process by doing the following. Use the left arrow key to move the cursor close to the intersection point on the left. Press the Enter key three times.

Notice that y = 0, and (-1.192582,0) is an approximate x-intercept. Go through the same process to find the other x-intercept. Press 2nd|Calc|Intersect. Use the right arrow to move the cursor to the right intercept … Press the Enter key three times.

Here we find that y = 0, and (4 Here we find that y = 0, and (4.1925824,0) is an approximate x-intercept. The two intercepts are (-1.192582,0) and (4.1925824,0) Rounded to the nearest thousandth, the two solutions to the quadratic equation are: x = -1.193, 4.193

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