Quadratic Forms and Their Graphs

7.10.2 Quadratic Forms and Their Graphs

7.10.2 • Quadratic Forms and Their Graphs

Activity

In an earlier lesson, we saw that an equation such as $h (t) = 10 + 78 t - 16 t^{2}$ can model the height of an object thrown upward from a height of 10 feet with a vertical velocity of 78 feet per second.

A young man in a purple sweater stands on railroad tracks in a forest, looking up as he juggles or catches a red apple above his hands. Lush green trees frame the scene.

1. Is the expression $- 16 t^{2} + 78 t + 10$ written in standard form? Be prepared to show your reasoning.

Compare your answer:

Yes. It is a sum of a squared term, a linear term, and a constant term. Written in the form $a x^{2} + b x + c$ , – 16 is $a$ , 78 is $b$ , and 10 is $c$ .

2. Paola said that the equation $g (t) = - 2 (8 t + 1) (t - 5)$ also defines the same function, written in factored form. Is Paola correct? Be prepared to show your reasoning.

Compare your answer:

Yes. Distribute the – 2 to $(8 t + 1)$ and you have $(- 16 t - 2)$ . Multiply this binomial by $(t - 5)$ . The result of $(- 16 t - 2) (t - 5)$ is $- 16 t^{2} + 78 t + 10$ .

3. Here is a graph representing both $g (t) = - 2 (8 t + 1) (t - 5)$ and $h (t) = 10 + 78 t - 16 t^{2}$ .

A parabola on a coordinate grid. The x-axis scale is 1 and extends from 0 to 6. The y-axis scale is 10 extends from 0 to 140.

a. What is the $y$ -value of the $y$ -intercept?

The $y$ -intercept is $(0, 10)$ .

b. What is the $x$ -value of the $x$ -intercept?

The $x$ -intercept is $(5, 0)$ .

c. What do each of these points mean in this situation?

Compare your answer:

The $y$ -intercept shows the height, in feet, of the object when it was thrown, and the $x$ -intercept shows the number of seconds that passed after the object was thrown before it hit the ground.

Self Check

The quadratic function $y=-(x-2)(x+3)$ is graphed below.

What are all of the $x$ -intercepts?

$x =6$
$x = - 3$ , $x = 2$
$x = 2, x = 6$
$x = - 3$

Additional Resources

Characteristics of Quadratic Graphs

The graph of $y = (x + 1) (x - 2)$ is shown below.

What are the $x$ - and $y$ -intercepts?

A parabola on a coordinate grid. The x-axis scale is 5 and extends from approximately negative 10 to 10. The y-axis scale is 5 and extends from approximately negative 10 to 12.

The $x$ -intercepts are where the graph crosses the $x$ -axis.

This happens at $(- 1, 0)$ and $(2, 0)$ .

The $y$ -intercept is where the graph crosses the $y$ -axis.

This happens at $(0, 2)$ .

Try it

Characteristics of Quadratic Graphs

The graph of $y = - (x - 4) (x + 1)$ is below.

What are the $x$ - and $y$ -intercepts of the graph?

Compare your answer:

Here is how to find the intercepts:

For the $x$ -intercepts: Look at where the graph crosses the $x$ -axis.

This happens at $x = - 1$ and $x = 4$ . So the points are $(- 1, 0)$ and $(4, 0)$ .

For the $y$ -intercept: Look at where the graph crosses the $y$ -axis.

This happens at $y = 4$ so the point is $(0, 4)$ .