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.