Question

Which statements are true about the graph of the function f(x) = x2 – 8x + 5? Select three options.

The function in vertex form is f(x) = (x – 4)2 – 11.
The vertex of the function is (–8, 5).
The axis of symmetry is x = 5.
The y-intercept of the function is (0, 5).
The function crosses the x-axis twice.

Answer 1

To analyze the function \( f(x) = x^2 - 8x + 5 \), let's determine the true statements one by one.

1. **Vertex Form**: The vertex form of a quadratic is given by \( f(x) = a(x - h)^2 + k \), where \((h, k)\) is the vertex. First, we can complete the square to rewrite the function in vertex form.

\[
f(x) = x^2 - 8x + 5
\]
To complete the square:
\[
f(x) = (x^2 - 8x + 16) - 16 + 5 = (x - 4)^2 - 11
\]
Thus, the function in vertex form is \( f(x) = (x - 4)^2 - 11 \). This statement is **true**.

2. **Vertex of the Function**: From the vertex form we found, the vertex is \((h, k) = (4, -11)\). Therefore, the vertex is not \((-8, 5)\). This statement is **false**.

3. **Axis of Symmetry**: The axis of symmetry can be found from the vertex, which is \(x = h\). From the vertex \((4, -11)\), the axis of symmetry is \(x = 4\). This statement is **false** since it is not \(x = 5\).

4. **Y-Intercept**: The y-intercept occurs when \(x = 0\). We can find the y-intercept by calculating \(f(0)\):
\[
f(0) = (0)^2 - 8(0) + 5 = 5
\]
Thus, the y-intercept is \((0, 5)\). This statement is **true**.

5. **X-Intercepts**: The function crosses the x-axis where \(f(x) = 0\). We can find the roots with the quadratic formula:
\[
x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} = \frac{8 \pm \sqrt{(-8)^2 - 4(1)(5)}}{2(1)} = \frac{8 \pm \sqrt{64 - 20}}{2} = \frac{8 \pm \sqrt{44}}{2} = \frac{8 \pm 2\sqrt{11}}{2} = 4 \pm \sqrt{11}
\]
Since there are two distinct real solutions, the function does indeed cross the x-axis twice. This statement is **true**.

In summary, the true statements about the function \( f(x) = x^2 - 8x + 5 \) are:

1. The function in vertex form is \( f(x) = (x - 4)^2 - 11 \).
4. The y-intercept of the function is \((0, 5)\).
5. The function crosses the x-axis twice.

So the three options that are true are 1, 4, and 5.