Question

Part A: What type of function is this?
Answer:

Part B: What transformations on the parent function have occurred?
Answer:

Part C: Create an equation for the function shown on the graph.
Answer:


the graph is starting from 0-4 on the x-axis and 0-5 on y axis with the Quadratic line formed over -3 of the x-axis

Answer 1

To answer your questions, let's analyze the information you've provided step by step.

### Part A: What type of function is this?
**Answer:** The function described is a **quadratic function**.

### Part B: What transformations on the parent function have occurred?
**Answer:** The parent quadratic function is \( f(x) = x^2 \). Based on your description that the graph starts from (0, 0) to (4, 5) and that it has a vertex shifted from \( (0, 0) \) to \( (-3, 0) \), it suggests the function has undergone the following transformations:
- **Horizontal Shift:** The graph has shifted to the left by 3 units, indicating a transformation in the form \( f(x) = (x + 3)^2 \).
- **Vertical Stretch/Compression:** If the peak is at a specific value along the y-axis between (0, 0) and (4, 5), then there may be a vertical stretch or compression based on the exact peak height.

Assuming no additional vertical stretch or compression, the transformation can simply be described as a leftward shift.

### Part C: Create an equation for the function shown on the graph.
**Answer:** Considering the vertex form of a quadratic equation and the information about the vertex being at \( (-3, 0) \):

The equation can be formulated as:
\[
f(x) = a(x + 3)^2 + k
\]
where \( a \) is a coefficient that determines the width and the direction of the parabola (upward or downward) and \( k \) would adjust the vertical position.

Assuming the vertex is at \( (-3, 0) \) (which we already identified), and if we want the parabola to open upwards from there, we could initially let \( k = 0 \).

To find the value of \( a \), we might consider a point the parabola passes through. If you are comfortable with the maximum height at \( (0, 4) \), we can deduce a specific \( a \) from any point along the curve.

For instance, applying \( (0, 4) \):
\[
4 = a(0 + 3)^2 + 0 \\
4 = 9a \\
a = \frac{4}{9}
\]

Thus, the equation could be:
\[
f(x) = \frac{4}{9}(x + 3)^2
\]

This assumes the only transformation was a horizontal shift and that the vertex remains at \( ( -3, 0) \). Please validate this by observing the specific points or shape of the graph you are analyzing.