Question

Graph the function.
​f(x)equalsleft brace Start 2 By 2 Matrix 1st Row 1st Column negative 3 x squared comma 2nd Column for x less than or equals 0 2nd Row 1st Column one half x plus 3 comma 2nd Column for x greater than 0 EndMatrix

Question content area bottom
Part 1
Choose the correct graph below.
A.
-10
10
-14
10
x
y



x y graph
B.
-10
10
-14
10
x
y



x y graph
C.
-10
10
-14
10
x
y



x y graph
D.
-10
10
-14
10
x
y

Answer 1

To graph the piecewise function

\[
f(x) =
\begin{cases}
-3x^2 & \text{for } x \leq 0 \\
\frac{1}{2}x + 3 & \text{for } x > 0
\end{cases}
\]

we need to analyze and graph both pieces of the function separately.

1. **For \(x \leq 0\):** The function is defined as \(f(x) = -3x^2\).
- This is a downward-opening parabola.
- The vertex of the parabola is at the origin (0, 0).
- When \(x = 0\), \(f(0) = 0\).
- As \(x\) approaches negative infinity, \(f(x)\) approaches negative infinity.

2. **For \(x > 0\):** The function is defined as \(f(x) = \frac{1}{2}x + 3\).
- This is a linear function with a slope of \(\frac{1}{2}\) and a y-intercept of 3.
- When \(x = 0\), the limit approaches \(f(0) = 3\) (but since it's only for \(x > 0\), the point (0, 3) is not included in the graph).
- When \(x = 2\), \(f(2) = \frac{1}{2}(2) + 3 = 4\).
- When \(x = 4\), \(f(4) = \frac{1}{2}(4) + 3 = 5\).
- The line will continue to rise as \(x\) increases.

### Steps to Create the Graph:

- Plot the downward-opening parabola \(f(x) = -3x^2\) for \(x \leq 0\) which includes points like (0,0), (-1, -3), (-2, -12), etc.
- For \(x > 0\), plot the line \(f(x) = \frac{1}{2}x + 3\) which starts just above (0, 3) (open circle at (0, 3)), passing through (2, 4), and (4, 5), etc.
- The graph should show a clear transition at the point \(x = 0\) from the parabola to the line.

### Final Selection
Without access to the graphs provided in the question, I can't select an exact option. However, based on the described behavior of the function, look for:

- A parabola that opens downwards on the left side (where \(x \leq 0\)).
- A linear line with a positive slope beginning right above (0, 3) on the right side (where \(x > 0\)) and continuing upwards.

Check the available options for the one that best matches these characteristics.