Question

A quadratic function is shown. f(x)=3x2−12x+11
. Complete the statements.(4 points)
Put responses in the correct input to answer the question. Select a response, navigate to the desired input and insert the response. Responses can be selected and inserted using the space bar, enter key, left mouse button or touchpad. Responses can also be moved by dragging with a mouse.
The function f(x)=3x2−12x+11
has a vertex of Response area and Response area is the equation of the axis of symmetry. The y-intercept of the function is Response area and one of the zeros is Response area rounded to the nearest tenth.

Answer 1

To complete the statements about the quadratic function \( f(x) = 3x^2 - 12x + 11 \), we need to find the following:

1. **Vertex**: The vertex of a quadratic function \( ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \). For this function:

- \( a = 3 \)
- \( b = -12 \)

Calculating the x-coordinate of the vertex:
\[
x = -\frac{-12}{2 \cdot 3} = \frac{12}{6} = 2
\]
To find the y-coordinate, substitute \( x = 2 \) into \( f(x) \):
\[
f(2) = 3(2^2) - 12(2) + 11 = 3(4) - 24 + 11 = 12 - 24 + 11 = -1
\]
Thus, the vertex is \( (2, -1) \).

2. **Axis of Symmetry**: This is the vertical line that passes through the vertex, given by the equation \( x = 2 \).

3. **Y-Intercept**: This is found by evaluating \( f(0) \):
\[
f(0) = 3(0^2) - 12(0) + 11 = 11
\]
Therefore, the y-intercept is \( 11 \).

4. **Zeros**: To find the zeros of the function, set \( f(x) = 0 \):
\[
3x^2 - 12x + 11 = 0
\]
We can use the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
With \( a = 3 \), \( b = -12 \), and \( c = 11 \):
\[
x = \frac{12 \pm \sqrt{(-12)^2 - 4 \cdot 3 \cdot 11}}{2 \cdot 3} = \frac{12 \pm \sqrt{144 - 132}}{6} = \frac{12 \pm \sqrt{12}}{6} = \frac{12 \pm 2\sqrt{3}}{6} = \frac{6 \pm \sqrt{3}}{3}
\]
The two zeros are:
\[
x_1 = 2 + \frac{\sqrt{3}}{3} \quad \text{and} \quad x_2 = 2 - \frac{\sqrt{3}}{3}
\]
Approximating \( \sqrt{3} \approx 1.732 \):
\[
x_1 \approx 2 + 0.577 \approx 2.577, \quad x_2 \approx 2 - 0.577 \approx 1.423
\]
Rounding \( x_2 \) to the nearest tenth gives \( 1.4 \).

### Final Responses:

- The function \( f(x) = 3x^2 - 12x + 11 \) has a vertex of **(2, -1)** and **x = 2** is the equation of the axis of symmetry.
- The y-intercept of the function is **11** and one of the zeros is **1.4** rounded to the nearest tenth.