algebra 8 5 study guide solving quadratic equations by...

Algebra 8-5 Study Guide: Solving Quadratic Equations by Graphing (pp 554-555) Page ! of !1 12

Attendance Problems 1. Graph y = x2 + 4x + 3.

2. What are the vertex and the zeros?


• Solve quadratic equations by graphing.

Vocabulary: Quadratic equation

Common Core CC.9-12.A.REI.11 Explain why the x- coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, e.g., using technology to graph the functions, make tables of values,….* CC.9-12.A.REI.4 Solve quadratic equations by inspection (e.g., for x2 = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial from of the equation. Recognize when the quadratic formula gives complex solutions and write them as a ± bi for real numbers a and b. CC.9-12.F.IF.7 Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.*

a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable factorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

3. What is a quadratic equation?


4. What is the standard form of a quadratic equation?

Question: What does a person have if they can easily find solutions to quadratic equations? Answer: Zero visibility.

"You gain strength, courage and confidence by every experience in which you really stop to look fear in the face.”—Teddy Roosevelt

One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros.


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Solving Quadratic Equationsby Graphing

ObjectiveSolve quadratic equations by graphing.

Vocabularyquadratic equation

Every quadratic function has a related quadratic equation. A quadratic equation is an equation that can be written in the standard form a x 2 + bx + c = 0,where a, b, and c are real numbers and a ≠ 0.

When writing a quadratic function as its related quadratic equation, you replace y with 0.

y = a x 2 + bx + c 0 = a x 2 + bx + c

ax2 + bx + c = 0

One way to solve a quadratic equation in standard form is to graph the related function and find the x-values where y = 0. In other words, find the zeros of the related function. Recall that a quadratic function may have two, one, or no zeros.

Solving Quadratic Equations by Graphing

Step 1 Write the related function.

Step 2 Graph the related function.

Step 3 Find the zeros of the related function.

1E X A M P L E Solving Quadratic Equations by Graphing

Solve each equation by graphing the related function.

A 2 x 2 - 2 = 0Step 1 Write the related function.

2 x 2 - 2 = y, or y = 2x2 + 0x - 2Step 2 Graph the function.

• The axis of symmetry is x = 0.• The vertex is (0, -2) .• Two other points are (1, 0) and (2, 6) .• Graph the points and reflect them across

the axis of symmetry.Step 3 Find the zeros.

The zeros appear to be -1 and 1.

Check 2 x 2 - 2 = 0 2 x 2 - 2 = 0 2 (-1) 2 - 2 0 Substitute -1 and 1 for x

in the quadratic equation. 2 (1) 2 - 2 0

2 (1) - 2 0 2 (1) - 2 0 2 - 2 0 2 - 2 0 0 0 ✓ 0 0 ✓

Who uses this?Dolphin trainers can use solutions of quadratic equations to plan the choreography for their shows. (See Example 2.)

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554 Chapter 8 Quadratic Functions and Equations

8-5CC.9-12.A.REI.11 Explain why the x- coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions approximately, ….* Also CC.9-12.A.REI.4, CC.9-12.F.IF.7*, CC.9-12.F.IF.4*

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Solve each equation by graphing the related function.

B - x 2 - 4x - 4 = 0

Step 1 Write the related function.y = - x 2 - 4x - 4

Step 2 Graph the function.

• The axis of symmetry is x = -2.

• The vertex is (-2, 0) .

• The y-intercept is -4.

• Another point is (-1, -1) .

• Graph the points and reflect them across the axis of symmetry.

Step 3 Find the zeros.The only zero appears to be -2.

Check - x 2 - 4x - 4 = 0 - (-2) 2 -4 (-2) - 4 0 - (4) + 8 - 4 0 4 - 4 0 0 0 ✓

You can also confirm the solution by using the Table function. Enter the

equation and press . When y = 0, x = -2. The x-intercept is -2.

C x 2 + 5 = 4x

Step 1 Write the related function. x 2 - 4x + 5 = 0y = x 2 - 4x + 5

Step 2 Graph the function. Use a graphing calculator.

Step 3 Find the zeros. The function appears to have no zeros.

The equation has no real-number solutions.

Check reasonableness Use the table function.

There are no zeros in the Y1 column.Also, the signs of the values in thiscolumn do not change. The functionappears to have no zeros.

Solve each equation by graphing the related function. 1a. x 2 - 8x - 16 = 2 x 2 1b. 6x + 10 = - x 2 1c. - x 2 + 4 = 0

8-5 Solving Quadratic Equations by Graphing 555

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Example 1. Solve the equation by graphing the related function. A. ! B. ! C. ! 2x2 −18 = 0 −12x +18 = −2x2 2x2 + 4x = −3


Guided Practice: Solve the equation by graphing the related function. 5. x2 – 8x – 16 = 2x2 6. 6x + 10 = –x2


7. –x2 + 4 = 0


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2E X A M P L E Aquatics Application

A dolphin jumps out of the water. The quadratic function y = -16 x 2 + 20x models the dolphin’s height above the water after x seconds. How long is the dolphin out of the water?

When the dolphin leaves the water, its height is 0, and when the dolphin reenters the water, its height is 0. So solve 0 = - 16x 2 + 20x to find the times when the dolphin leaves and reenters the water.

Step 1 Write the related function.0 = -16 x 2 + 20xy = - 16x 2 + 20x

Step 2 Graph the function.Use a graphing calculator.

Step 3 Use to estimate the zeros.The zeros appear to be 0 and 1.25.The dolphin leaves the water at 0 seconds and reenters the water at 1.25 seconds.

The dolphin is out of the water for 1.25 seconds.

Check 0 = -16 x 2 + 20x

0 -16 (1.25) 2 + 20 (1.25)

0 -16 (1.5625) + 25

0 -25 + 25

0 0 ✓

Substitute 1.25 for x in the

quadratic equation.

2. What if…? Another dolphin jumps out of the water. The quadratic function y = -16 x 2 + 32 x models the dolphin’s height above the water after x seconds. How long is the dolphin out of the water?

THINK AND DISCUSS 1. Describe the graph of a quadratic function whose related quadratic

equation has only one solution.

2. Describe the graph of a quadratic function whose related quadratic equation has no real solutions.

3. Describe the graph of a quadratic function whose related quadratic equation has two solutions.

4. GET ORGANIZED Copy and complete the graphic organizer. In each of the boxes, write the steps for solving quadratic equations by graphing.

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556 Chapter 8 Quadratic Functions and Equations

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Example 2. A frog jumps straight up from the ground. The quadratic function f(t) = –16t2 + 12t models the frog’s height above the ground after t seconds. About how long is the frog in the air?


8. Guided Practice: A dolphin jumps out of the water. The quadratic function y = –16x2 + 32x models the dolphin’s height above the water after x seconds. How long is the dolphin out of the water?


8-5 Solving Quadratic Equations by Graphing: (p 557-558) 17, 19, 23-30, 38-40.

Question: What does a person have if they can easily find solutions to quadratic equations? Answer: Zero visibility.

algebra 8 5 study guide solving quadratic equations by...

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