Match Quadratic Graphs and Their Equations: Mini-Lesson Review

Mini Lesson Question

Question #3: Match Quadratic Graphs and Their Equations

Examine the graph below.

$GRAPH OF A PARABOLA THAT OPENS UPWARD WITH A $y$-intercepts OF NEGATIVE 3 AND $x$-intercepts OF NEGATIVE 1 AND 3.$

Which function matches the graph?

$f(x) = (x + 1)(x + 3)$
$f(x) = (x - 1)(x - 3)$
$f(x) = (x +1)(x - 3)$
$f(x) = (x - 1)(x + 3)$

Match Quadratic Graphs and Their Equations

The factored form helps find the zeros of a quadratic function.

The factored form gives the zeros, and the $x$ -intercepts are located where each factor equals 0.

Example

Examine the graph. What are the $x$ -intercepts of the graph?

Graph of a parabola that opens upward with a y-intercepts of negative 4 and x-intercepts of negative 4 and 1.

The graph crosses the $x$ -axis at $x = - 4$ and $x = 1$ . These are the $x$ -intercepts.

Look at each function to see which matches the graph. Look for the factors that have zeros as $x = - 4$ and $x = 1$ .

a. $f (x) = (x + 4) (x + 1)$

This function has zeros at $x = - 4$ and $x = - 1$ . The zero $x = - 1$ is not a zero on the graph.

b. $f (x) = (x - 4) (x - 1)$

This function has zeros at $x = 4$ and $x = 1$ . The zero $x = 4$ is not a zero on the graph.

c. $f (x) = (x - 4) (x + 1)$

This function has zeros at $x = 4$ and $x = - 1$ . The signs of the zeros of this function are the opposite of the signs of the zeros on the graph.

d. $f (x) = (x + 4) (x - 1)$

This function has zeros at $x = - 4$ and $x = - 1$ . This function has zeros that match the zeros on the graph.

The function $f (x) = (x + 4) (x - 1)$ matches the graph.

Try it

Match Quadratic Graphs and Their Equations

Examine the graph.

Graph of a parabola that opens upward with a y-intercepts of negative 8 and x-intercepts of negative 2 and 4.

Which function matches the graph?

a. $f (x) = (x + 2) (x + 4)$

b. $f (x) = (x - 2) (x - 4)$

c. $f (x) = (x + 2) (x - 4)$

d. $f (x) = (x - 2) (x + 4)$

Here’s how to match the function to the graph.

Look at each function to see which matches the graph. Look for the factors that have zeros at $x = - 2$ and $x = 4$ .

a. $f (x) = (x + 2) (x + 4)$

This function has zeros at $x = - 2$ and $x = - 4$ . The zero $x = - 4$ is not a zero on the graph.

b. $f (x) = (x - 2) (x - 4)$

This function has zeros at $x = 2$ and $x = 4$ . The zero $x = 2$ is not a zero on the graph.

c. $f (x) = (x + 2) (x - 4)$

This function has zeros at $x = - 2$ and $x = 4$ . This function has zeros that match the zeros on the graph.

d. $f (x) = (x - 2) (x + 4)$

This function has zeros at $x = 2$ and $x = - 4$ . The zeros of the function have the opposite signs of the zeros on the graph.

The function $f (x) = (x + 2) (x - 4)$ matches the graph.

Check Your Understanding

Examine the graph.

Graph of a parabola that opens upward with a y-intercepts of negative 6 and x-intercepts of negative 3 and 2.

Which function matches the graph?

$f (x) = (x - 3) (x - 2)$
$f (x) = (x - 3) (x + 2)$
$f (x) = (x + 3) (x - 2)$
$f (x) = (x + 3) (x + 2)$

$f (x) = (x + 3) (x - 2)$

Check yourself: Looking at the graph, the parabola has zeros at $x = - 3$ and $x = 2$ . We can use the zero pairs of these values to determine the binomial factors for the $x + 3$ function: and $x - 2$ . The function $f (x) = (x + 3) (x - 2)$ has zeros at $x = - 3$ and $x = 2$ .