Representations of Quadratic Functions

7.12.4 Representations of Quadratic Functions

7.12.4 • Representations of Quadratic Functions

Activity

Your teacher will give your group a set of cards. Each card contains a graph or an equation.

Take turns with your partner to sort the cards into sets so that each set contains two equations and a graph that all represent the same quadratic function.
For each set of cards that you put together, explain to your partner how you know they belong together.
For each set that your partner puts together, listen carefully to their explanation. If you disagree, discuss your thinking and work to reach an agreement.

Once all the cards are sorted and discussed, record the equivalent equations, sketch the corresponding graph, and write a brief note or explanation about why the representations were grouped together.

Compare your answer:

Set 1: Standard Form: $y = x^{2} - 1$ Factored Form: $y = (x + 1) (x - 1)$

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 4 to 6. The y-axis scale is 1 and extends from negative 6 to 7.

Set 2: Standard Form: $y = x^{2} - 4 x$ Factored Form: $y = x (x - 4)$

Set 3: Standard Form: $y = x^{2} - 5 x + 4$ Factored Form: $y = (x - 1) (x - 4)$

Set 4: Standard Form: $y = x^{2} - 4 x + 4$ Factored Form: $y = (x - 2)^{2}$

Self Check

Examine the graph below.

$GRAPH OF A PARABOLA THAT OPENS UPWARD WITH A $y$-intercepts OF NEGATIVE 2 AND $x$-intercepts OF NEGATIVE 2 AND 1.$

Which function matches the graph?

$f(x) = (x - 2)(x - 1)$
$f(x) = (x + 2)(x + 1)$
$f(x) = (x + 2)(x - 1)$
$f(x) = (x - 2)(x + 1)$

Additional Resources

Matching Quadratic Functions and Their Graphs

Below is a quadratic function represented in standard form, factored form, and with a graph.

Standard Form: $f (x) = x^{2} + 2 x - 8$

Factored Form: $f (x) = (x + 4) (x - 2)$

A parabola on a coordinate grid. The x-axis scale is 1 and extends from negative 10 to 10. The y-axis scale is 1 and extends from negative 14 to 6.

Notice the factored form helps find the zeros: $x = - 4$ and $x = 2$ .

The factored form gives the zeros since the zeros ( $x$ -intercepts) are located where each factor equals 0.

The standard form, $f (x) = x^{2} + 2 x - 8$ , helps give the $y$ -intercept. The $y$ -intercept occurs when $x = 0$ . Here, $f (0) = - 8$ . On the graph, the $y$ -intercept is $(0, - 8)$ .

Try it

Matching Quadratic Functions and Their Graphs

Given the graph below and its function in standard form and factored form, identify all intercepts.

Standard form: $f (x) = x^{2} - x - 6$ Factored form: $f (x) = (x - 3) (x + 2)$

Here is how to find the intercepts with the information provided:

The graph and the factored form help give the $x$ -intercepts. They occur where the factors of the factored form equal 0, at $x = - 2$ and $x = 3$ . On the graph, these are where the graph crosses the $x$ -axis, at the points $(- 2, 0)$ and $(3, 0)$ .

The $y$ -intercept can be found with standard form when $x = 0$ in $f (x) = x^{2} - x - 6$ . So $y = - 6$ . Notice on the graph, the $y$ -intercept, where the graph crosses the $y$ -axis, is at $(0, - 6)$ .