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38.	In the figure, P is the vertex of the graph of the quadratic function
y = a(x – h)2 + 17.
(a)	Find the values of a and h.
(b)	Find the y-intercept of the graph.	


Let's solve the given problem step by step using the information provided in the image and the given quadratic function.

The quadratic function is given as:
\[ y = a(x - h)^2 + 17 \]

### (a) Find the values of \(a\) and \(h\)

1. **Vertex of the Parabola**:
   The vertex form of a quadratic function is \( y = a(x - h)^2 + k \). Here, the vertex \( P \) is given as \((-1, 5)\).
   From the given equation, we see that \( k = 17 \). This should match the y-coordinate of the vertex, but from the graph, the y-coordinate of the vertex \( P \) is 5. Thus, there is a mistake in the provided equation or a misunderstanding of the graph.

2. **Matching Vertex Coordinates**:
   Given the vertex \((-1, 5)\), we have:
   \[ y = a(x + 1)^2 + 5 \]
   This corrects the form to match the vertex given in the graph.

3. **Substitute the Point**:
   We have a point on the graph \((-3, 0)\).
   Substitute \(x = -3\) and \(y = 0\) into the equation:
   \[ 0 = a(-3 + 1)^2 + 5 \]
   \[ 0 = a(-2)^2 + 5 \]
   \[ 0 = 4a + 5 \]
   \[ 4a = -5 \]
   \[ a = -\frac{5}{4} \]

So, the values of \(a\) and \(h\) are:
\[ a = -\frac{5}{4} \]
\[ h = -1 \]

### (b) Find the y-intercept of the graph

To find the y-intercept, we set \( x = 0 \) in the equation:
\[ y = -\frac{5}{4}(0 + 1)^2 + 5 \]
\[ y = -\frac{5}{4} \cdot 1 + 5 \]
\[ y = -\frac{5}{4} + 5 \]
\[ y = -\frac{5}{4} + \frac{20}{4} \]
\[ y = \frac{15}{4} \]

So, the y-intercept of the graph is:
\[ y = \frac{15}{4} \]

If you have any questions or need further details, feel free to ask. Here are some questions you might want to explore next:

1. How do we derive the general form of a quadratic equation from its vertex form?
2. What are the properties of the vertex of a parabola?
3. How does the value of \(a\) affect the direction and shape of the parabola?
4. Can we find the x-intercepts of this parabola?
5. How do we complete the square to convert a standard quadratic equation to vertex form?

**Tip:** When analyzing a quadratic function, always identify key features like the vertex, axis of symmetry, and intercepts to understand the graph's shape and position.

Learn how to solve the quadratic function y = a(x – h)2 + 17 step-by-step. Find the values of a and h from the vertex form and calculate the y-intercept of the graph. This problem involves identifying the vertex, applying the vertex form formula, and understanding quadratic functions.

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