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

$$
LATEXBLOCK0
$$

Step 2: Convert Vertex Form to Intercept Form

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

$$
LATEXBLOCK1
$$

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

$$
LATEXBLOCK2
$$

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:

$$
LATEXBLOCK3
$$

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:

$$
LATEXBLOCK4
$$

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

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

$$
LATEXBLOCK5
$$

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

$$
LATEXBLOCK6
$$

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:

$$
LATEXBLOCK7
$$

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:

$$
LATEXBLOCK8
$$

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

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

$$
LATEXBLOCK9
$$

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