Question

Which is the graph of the function f(x) = one-halfx2 + 2x – 6?

Answer 1

To graph the function \( f(x) = \frac{1}{2}x^2 + 2x - 6 \), we can follow these steps:

1. **Identify the type of function**: This is a quadratic function, which will graph as a parabola. Since the coefficient of \( x^2 \) (which is \( \frac{1}{2} \)) is positive, the parabola opens upwards.

2. **Find the vertex**: The vertex of a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula:
\[
x = -\frac{b}{2a}
\]
Here, \( a = \frac{1}{2} \) and \( b = 2 \):
\[
x = -\frac{2}{2 \cdot \frac{1}{2}} = -2
\]
Then find \( f(-2) \):
\[
f(-2) = \frac{1}{2}(-2)^2 + 2(-2) - 6 = \frac{1}{2}(4) - 4 - 6 = 2 - 4 - 6 = -8
\]
So, the vertex is at the point \( (-2, -8) \).

3. **Find the y-intercept**: To find the y-intercept, evaluate \( f(0) \):
\[
f(0) = \frac{1}{2}(0)^2 + 2(0) - 6 = -6
\]
So, the y-intercept is \( (0, -6) \).

4. **Find the x-intercepts**: Set \( f(x) = 0 \) to find the x-intercepts:
\[
\frac{1}{2}x^2 + 2x - 6 = 0
\]
Multiply through by 2 to eliminate the fraction:
\[
x^2 + 4x - 12 = 0
\]
Now, factor the quadratic:
\[
(x + 6)(x - 2) = 0
\]
This gives the solutions \( x = -6 \) and \( x = 2 \). Thus, the x-intercepts are \( (-6, 0) \) and \( (2, 0) \).

5. **Plot the points**: Plot the vertex \( (-2, -8) \), the y-intercept \( (0, -6) \), and the x-intercepts \( (-6, 0) \) and \( (2, 0) \).

6. **Draw the parabola**: Connect the points smoothly to form the parabola that opens upwards.

The graph will have the vertex at \( (-2, -8) \), will cross the y-axis at \( (0, -6) \) and the x-axis at \( (-6, 0) \) and \( (2, 0) \). This will create a U-shaped curve typical of quadratic functions.

If you need a visual representation, you can use graphing software or a calculator to plot the function.