Changing the Parameters of a Quadratic Expression

7.17.3 Changing the Parameters of a Quadratic Expression

7.17.3 • Changing the Parameters of a Quadratic Expression

Activity

Mai is learning to create computer animation by programming. In one part of her animation, she uses a quadratic function to model the path of the main character, an animated peanut, jumping over a wall.

Two peanuts with wire arms and legs stand on a purple background, resembling a parent and child holding hands, with the larger peanut gently holding the smaller ones hand.

Mai uses the equation $f (x) = - 0.1 (x - h)^{2} + k$ to represent the path of the jump. $f$ represents the height of the peanut as a function of the horizontal distance it travels, $x$ .

On the screen, the base of the wall is located at $(22, 0)$ , with the top of the wall at $(22, 4.5)$ . The dashed curve in the picture shows the graph of one equation Mai tried, where the peanut fails to make it over the wall.

Graph of a parabola on a coordinate plane. The x-axis has a scale of 2 and extends from 0 to 30. The y-axis has a scale of 1 and extends from 0 to 8.

What is the value of $h$ in this equation?

$h = 18$ .

What is the value of $k$ in this equation?

$k = 5$ .

Starting with Mai's equation, choose values for $h$ and $k$ that will guarantee the peanut stays on the screen but also makes it over the wall. Be prepared to explain your reasoning.

Compare your answer:

Use 22 for $h$ . Because $h$ is the $x$ -coordinate of the highest point of the jump, if that point is directly over the wall and the $k$ -value remains at 5, then the peanut would clear the wall. So the new equation would be $f (x) = - 0.1 (x - 22)^{2} + 5$ .
Change the value of $k$ , which is the vertical coordinate of the vertex, to 7 so that the vertex is at $(18, 7)$ and there is still enough vertical distance to clear the 4.5-unit-tall wall when $x$ is 22. The new equation would be $f (x) = - 0.1 (x - 18)^{2} + 7$ .

Video: Changing the Parameters of a Quadratic Expression

Watch the following video to learn more about changing the parameters of a quadratic expression.

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Self Check

Cash kicked a football off a tee. The path of the football is shown below and represented by the function $f(x)=-0 .02(x-h)^2+k$ , where $x$ represents the horizontal distance and $f(x)$ represents the height.

Find the vertex and its meaning.

$(30, 0)$ , the football travels 30 feet before hitting the ground.
$(14,5)$ the football hit its highest height of 14 feet after traveling 5 feet.
$(14,5)$ , the football hit its highest height of 5 feet after traveling 14 feet.
$(0,1)$ , the ball was kicked from a height of 1 foot.

Additional Resources

Determining the Vertex From a Graph

Zaren made a paper football for a game at indoor recess and flicked it in the air from a table. The path of the paper football is shown below and represented by the function $f (x) = - 0 .02 (x - h)^{2} + k$ .

Graph of a parabola on a coordinate plane. The x-axis represents the horizontal distance in feet and has a scale of 1 extending from 0 to approximately 36. The vertical-axis represents the height of the football in feet and extends from 0 to 10 with a scale of 1.

Find the value of the vertex and its meaning:

The peak of the graph is at $(17, 3)$ . This means the maximum horizontal distance traveled was 17 inches when the height of the object was 3 inches.

Try it

Determining the Vertex From a Graph

A parabola on a coordinate grid. The x-axis represents the ticket price in dollars and the y-axis represents the revenue in dollars. The x-axis scale is 5 and extends from 0 to 40. The y-axis scale is 200 extends from 0 to 1,400.

The revenue made from a high school play is shown in the graph above. Find the vertex and tell what it means in this situation.

Here is how to find the vertex and determine its meaning:

The peak of the graph is at $(15, 1100)$ . This means that the maximum revenue is $1100 when the ticket price is $15.