Question

function 1: h(x) = -(x - 2)² + 9; function 2: (image of graph of j(x))

Explanation:

Step1: Analyze Function 1

The function \( h(x) = -(x - 2)^2 + 9 \) is a quadratic function in vertex form \( y = a(x - h)^2 + k \), where the vertex is at \( (h, k)=(2, 9) \), and since \( a=-1<0 \), it opens downward. The roots (x - intercepts) can be found by setting \( h(x) = 0 \):

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So \( x = 5 \) or \( x=-1 \), so the x - intercepts are \( (-1, 0) \) and \( (5, 0) \), and the y - intercept is when \( x = 0 \): \( h(0)=-(0 - 2)^2+9=-4 + 9 = 5 \), so y - intercept is \( (0, 5) \).

Step2: Analyze Function 2 (Graph of \( j(x) \))

From the graph of \( j(x) \), we can see the vertex (the highest point) is at \( (2, 5) \) (by looking at the grid, the peak is at x = 2, y = 5). The x - intercepts: when y = 0, x=-1 and x = 5 (from the graph, the curve crosses the x - axis at \( (-1, 0) \) and \( (5, 0) \)), and the y - intercept is at \( (0, 5) \) (when x = 0, y = 5 from the graph).

Step3: Compare Key Features

• Vertex: For \( h(x) \), vertex is \( (2, 9) \); for \( j(x) \), vertex is \( (2, 5) \).
• y - intercept: For \( h(x) \), y - intercept is \( (0, 5) \); for \( j(x) \), y - intercept is \( (0, 5) \).
• x - intercepts: Both have x - intercepts at \( (-1, 0) \) and \( (5, 0) \).
• Direction of opening: Both open downward (since the coefficient of \( x^2 \) for \( h(x) \) is - 1 and the graph of \( j(x) \) opens downward as it has a maximum point).

If we were to find differences (e.g., vertex y - coordinate) or similarities, we can see that the x - intercepts and y - intercept (except for the vertex y - value) have some commonalities. If the question was about comparing the two functions (e.g., vertex, intercepts), we can summarize the key points as above.

Answer:

(If the question was about, for example, the x - intercepts of both functions, the x - intercepts of both \( h(x) \) and \( j(x) \) are \( \boldsymbol{(-1, 0)} \) and \( \boldsymbol{(5, 0)} \); if about vertex, \( h(x) \) has vertex \( (2, 9) \), \( j(x) \) has vertex \( (2, 5) \); if about y - intercept, both have y - intercept \( (0, 5) \) etc. Since the question is not fully specified, but based on the analysis, we can provide the key features as above. If we assume the question is to find the x - intercepts of both functions, the x - intercepts are \( (-1, 0) \) and \( (5, 0) \) for both.)