Question

algebra 2 - 2.1.2 skills practice b
name:
complete the table by converting the given quadratic function into its other two forms.
general form\tvertex form\tintercept form
\t$f(x) = 2(x - 3)^2 - 32$\t
\t\t$f(x) = \frac{1}{2}(x + 2)(x - 4)$
$f(x) = 5x^2 - 70x - 225$\t\t
\t\t$f(x) = -0.25(x - 4)(x + 6)$
$f(x) = -3x^2 + 36x - 81$\t\t

Answer:

Let's solve each row of the table by converting the quadratic function into the required forms. We'll start with the first row.

Row 1: Vertex Form given \( f(x) = 2(x - 3)^2 - 32 \)

Step 1: Convert Vertex Form to General Form

Expand \( 2(x - 3)^2 - 32 \):
\[

$$\begin{align*} f(x) &= 2(x^2 - 6x + 9) - 32 \\ &= 2x^2 - 12x + 18 - 32 \\ &= 2x^2 - 12x - 14 \end{align*}$$

\]

Step 2: Convert Vertex Form to Intercept Form

Set \( f(x) = 0 \):
\[

$$\begin{align*} 2(x - 3)^2 - 32 &= 0 \\ 2(x - 3)^2 &= 32 \\ (x - 3)^2 &= 16 \\ x - 3 &= \pm 4 \\ x &= 3 \pm 4 \end{align*}$$

\]
So the roots are \( x = 7 \) and \( x = -1 \). Then the intercept form is:
\[
f(x) = 2(x - 7)(x + 1)
\]

Row 2: Intercept Form given \( f(x) = \frac{1}{2}(x + 2)(x - 4) \)

Step 1: Convert Intercept Form to General Form

Expand \( \frac{1}{2}(x + 2)(x - 4) \):
\[

$$\begin{align*} f(x) &= \frac{1}{2}(x^2 - 4x + 2x - 8) \\ &= \frac{1}{2}(x^2 - 2x - 8) \\ &= \frac{1}{2}x^2 - x - 4 \end{align*}$$

\]

Step 2: Convert Intercept Form to Vertex Form

First, expand the intercept form to general form (already done above: \( f(x) = \frac{1}{2}x^2 - x - 4 \)). Then complete the square:
\[

$$\begin{align*} f(x) &= \frac{1}{2}(x^2 - 2x) - 4 \\ &= \frac{1}{2}(x^2 - 2x + 1 - 1) - 4 \\ &= \frac{1}{2}((x - 1)^2 - 1) - 4 \\ &= \frac{1}{2}(x - 1)^2 - \frac{1}{2} - 4 \\ &= \frac{1}{2}(x - 1)^2 - \frac{9}{2} \end{align*}$$

\]

Row 3: General Form given \( f(x) = 5x^2 - 70x - 225 \)

Step 1: Convert General Form to Vertex Form (Complete the Square)

Factor out the coefficient of \( x^2 \) from the first two terms:
\[

$$\begin{align*} f(x) &= 5(x^2 - 14x) - 225 \\ &= 5(x^2 - 14x + 49 - 49) - 225 \\ &= 5((x - 7)^2 - 49) - 225 \\ &= 5(x - 7)^2 - 245 - 225 \\ &= 5(x - 7)^2 - 470 \end{align*}$$

\]

Step 2: Convert General Form to Intercept Form (Factor)

Set \( f(x) = 0 \):
\[

$$\begin{align*} 5x^2 - 70x - 225 &= 0 \\ x^2 - 14x - 45 &= 0 \\ (x - 15)(x + 3) &= 0 \end{align*}$$

\]
So the roots are \( x = 15 \) and \( x = -3 \). Then the intercept form is:
\[
f(x) = 5(x - 15)(x + 3)
\]

Row 4: Intercept Form given \( f(x) = -0.25(x - 4)(x + 6) \)

Step 1: Convert Intercept Form to General Form

Expand \( -0.25(x - 4)(x + 6) \):
\[

$$\begin{align*} f(x) &= -0.25(x^2 + 6x - 4x - 24) \\ &= -0.25(x^2 + 2x - 24) \\ &= -0.25x^2 - 0.5x + 6 \end{align*}$$

\]

Step 2: Convert Intercept Form to Vertex Form

First, expand the intercept form to general form (already done above: \( f(x) = -0.25x^2 - 0.5x + 6 \)). Then complete the square:
\[

$$\begin{align*} f(x) &= -0.25(x^2 + 2x) + 6 \\ &= -0.25(x^2 + 2x + 1 - 1) + 6 \\ &= -0.25((x + 1)^2 - 1) + 6 \\ &= -0.25(x + 1)^2 + 0.25 + 6 \\ &= -0.25(x + 1)^2 + 6.25 \end{align*}$$

\]

Row 5: General Form given \( f(x) = -3x^2 + 36x - 81 \)

Step 1: Convert General Form to Vertex Form (Complete the Square)

Factor out the coefficient of \( x^2 \) from the first two terms:
\[

$$\begin{align*} f(x) &= -3(x^2 - 12x) - 81 \\ &= -3(x^2 - 12x + 36 - 36) - 81 \\ &= -3((x - 6)^2 - 36) - 81 \\ &= -3(x - 6)^2 + 108 - 81 \\ &= -3(x - 6)^2 + 27 \end{align*}$$

\]

Step 2: Convert General Form to Intercept Form (Factor)

Set \( f(x) = 0 \):
\[

$$\begin{align*} -3x^2 + 36x - 81 &= 0 \\ x^2 - 12x + 27 &= 0 \\ (x - 3)(x - 9) &= 0 \end{align*}$$

\]
So the roots are \( x = 3 \) and \( x = 9 \). Then the intercept form is:
\[
f(x) = -3(x - 3)(x - 9)
\]

Now, let's summarize the results in the table:

General Form	Vertex Form	Intercept Form
\( \frac{1}{2}x^2 - x - 4 \)	\( \frac{1}{2}(x - 1)^2 - \frac{9}{2} \)	\( \frac{1}{2}(x + 2)(x - 4) \)
\( 5x^2 - 70x - 225 \)	\( 5(x - 7)^2 - 470 \)	\( 5(x - 15)(x + 3) \)
\( -0.25x^2 - 0.5x + 6 \)	\( -0.25(x + 1)^2 + 6.25 \)	\( -0.25(x - 4)(x + 6) \)
\( -3x^2 + 36x - 81 \)	\( -3(x - 6)^2 + 27 \)	\( -3(x - 3)(x - 9) \)