Question

quadratic functions: standard form

what is the range of the function (f(x) = 3x^2 + 6x - 8)?

options:
\\(\\{y \mid y \ge -1\\}\\)
\\(\\{y \mid y \ge -11\\}\\)
\\(\\{y \mid y \le -11\\}\\)

Explanation:

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<reasoning>

Identify the quadratic coefficients

Using the Quadratic Functions knowledge point
\[

$$\begin{aligned} &f(x) = 3x^2 + 6x - 8\\ &a = 3,\quad b = 6,\quad c = -8 \end{aligned}$$

\]

Find the x-coordinate of the vertex

Using the Vertex of Quadratic Function knowledge point
\[

$$\begin{aligned} &x_v = -\frac{b}{2a} = -\frac{6}{2(3)} = -1 \end{aligned}$$

\]

Find the y-coordinate of the vertex

Using the Vertex of Quadratic Function knowledge point
\[

$$\begin{aligned} &y_v = f(-1) = 3(-1)^2 + 6(-1) - 8\\ &y_v = 3 - 6 - 8 = -11 \end{aligned}$$

\]

Determine the range of the function

Using the Domain and Range knowledge point
\[

$$\begin{aligned} &a = 3 > 0 \implies \text{parabola opens upward}\\ &\text{Range: } \{y \mid y \ge -11\} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>{y | y \(\ge\) -1}</mcq-option>
<mcq-correct>{y | y \(\ge\) -11}</mcq-correct>
<mcq-option>{y | y \(\le\) -11}</mcq-option>
<mcq-option>{y | y \(\le\) -1}</mcq-option>
</answer>

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Answer:

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<reasoning>

Identify the quadratic coefficients

Using the Quadratic Functions knowledge point
\[

$$\begin{aligned} &f(x) = 3x^2 + 6x - 8\\ &a = 3,\quad b = 6,\quad c = -8 \end{aligned}$$

\]

Find the x-coordinate of the vertex

Using the Vertex of Quadratic Function knowledge point
\[

$$\begin{aligned} &x_v = -\frac{b}{2a} = -\frac{6}{2(3)} = -1 \end{aligned}$$

\]

Find the y-coordinate of the vertex

Using the Vertex of Quadratic Function knowledge point
\[

$$\begin{aligned} &y_v = f(-1) = 3(-1)^2 + 6(-1) - 8\\ &y_v = 3 - 6 - 8 = -11 \end{aligned}$$

\]

Determine the range of the function

Using the Domain and Range knowledge point
\[

$$\begin{aligned} &a = 3 > 0 \implies \text{parabola opens upward}\\ &\text{Range: } \{y \mid y \ge -11\} \end{aligned}$$

\]
</reasoning>

<answer>
<mcq-option>{y | y \(\ge\) -1}</mcq-option>
<mcq-correct>{y | y \(\ge\) -11}</mcq-correct>
<mcq-option>{y | y \(\le\) -11}</mcq-option>
<mcq-option>{y | y \(\le\) -1}</mcq-option>
</answer>

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