7.17.1 Quadratic Expressions in Different Forms

Algebra 17.17.1 Quadratic Expressions in Different Forms

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Warm Up

Graphs of two parabolas on coordinate plane with unlabeled axes.

Here are graphs representing the functions $f$ and $g$ given by $f (x) = x (x + 6)$ and $g (x) = x (x + 6) + 4$ .

Which graph represents $f (x) = x (x + 6)$ ? How do you know?

Compare your answer:

The lower graph (which goes through the origin) is the graph of $f$ .

The value of $g$ is always 4 greater than the value of $f$ , no matter which value of $x$ we choose.
Rewriting the equation in standard form gives $f (x) = x^{2} + 6 x$ . This form shows that the $y$ -intercept of the graph representing $f$ is $(0, 0)$ .
The expression that defines function $f$ is in factored form. It shows that the two $x$ -intercepts are $(0, 0)$ and $(- 6, 0)$ .

Which graph represents $g (x) = x (x + 6) + 4$ ? How do you know?

Compare your answer:

Answer: The lower graph (which goes through the origin) is the graph of $f$ and the upper one is the graph of $g$ .

The value of $g$ is always 4 greater than the value of $f$ , no matter which value of $x$ we choose.
Rewriting the equation in standard form gives $g (x) = x^{2} + 6 x + 4$ .This form shows that the $y$ -intercept of the graph representing $g$ is $(0, 4)$ .

Where does the graph of $f$ meet the $x$ -axis? Explain how you know.

Compare your answer:

$x = - 6$ and $x = 0$ , because these are the two values of $x$ that make $x (x + 6)$ equal to 0.

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- Authors: Lynn Marecek, MaryAnne Anthony-Smith, Andrea Honeycutt Mathis, Jay Abramson, Sharon North, Amy Baldwin, Alyssa Howell
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- Book title: Algebra 1
- Publication date: Jun 4, 2025
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