Solving Quadratic Equations by Graphing · PDF fileSolving Quadratic Equations by Graphing Going Deeper Essential question: How can you solve a quadratic equation by graphing? Finding
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You can solve an equation of the form a(x - h ) 2 + k = c, which is called a quadratic equation, by graphing the functions f (x) = a(x - h ) 2 + k and g(x) = c and finding the
x-coordinate of each point of intersection.
Solving Quadratic Equations Graphically
Solve 2(x - 4 ) 2 + 1 = 7.
A Graph f (x) = 2(x - 4 ) 2 + 1.
What is the vertex?
If you move 1 unit right or left from the vertex, how
must you move vertically to be on the graph of f (x)?
What points are you at?
B Graph g(x) = 7.
C At how many points do the graphs of f (x) and g(x) intersect? If possible, find the
x-coordinate of each point of intersection exactly. Otherwise, give an approximation
of the x-coordinate of each point of intersection.
D For each x-value from part C, find the value of f (x). How does this show that you have
found actual or approximate solutions of 2(x - 4 ) 2 + 1 = 7?
REFLECT
2a. If you solved the equation 4(x - 3 ) 2 + 1 = 5 graphically, would you be able to
obtain exact or approximate solutions? Explain.
2b. For what value of c would the equation 4(x - 3 ) 2 + 1 = c have exactly one solution?
How is that solution related to the graph of f (x)?
5. Water is shot straight up out of a water soaker toy. The quadratic function y = −16x2 + 32x models the height in feet of a water droplet after x seconds. How long is the water droplet in the air?
2. The firework will explode when it reaches its highest point. How long after the fuse is lit will the firework explode and how high will the firework be?
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4. What is the meaning of each of the zeros you found in problem 3?
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5. The quadratic function ( ) = −16 2 + 90 models the height of a baseball in feet after seconds. How long is the baseball in the air? A 2.8125 s C 11.25 s B 5.625 s D 126.5625 s
7. The function = −0.04 2 + 2 models the height of an arch support for a bridge, where is the distance in feet from where the arch supports enter the water. How many real solutions does this function have? F 0 H 2 G 1 J 3
6. The height of a football in feet is given by the function = −16 2 + 56 + 2 where is the time in seconds after the ball was kicked. This function is graphed below. How long was the football in the air?
A 0.5 seconds C 2 seconds B 1.75 seconds D 3.5 seconds