Question

graph the parabola from the function given below.

\\g(x) = -2(x - 4)^2 + 7\\

Explanation:

⚡ Using what you learned: graphing quadratic functions

Step 1: Identify the vertex

The given quadratic function is in vertex form:

$$ g(x) = a(x - h)^2 + k $$

Comparing \( g(x) = -2(x - 4)^2 + 7 \) to the vertex form:

• \( a = -2 \)
• \( h = 4 \)
• \( k = 7 \)

The vertex of the parabola is at:

$$ (h, k) = (4, 7) $$

Step 2: Find additional points

Choose \( x \)-values near the vertex to find points on either side:

For \( x = 3 \):

$$ g(3) = -2(3 - 4)^2 + 7 = -2(-1)^2 + 7 = -2(1) + 7 = 5 $$

Point: \( (3, 5) \)

For \( x = 5 \):

$$ g(5) = -2(5 - 4)^2 + 7 = -2(1)^2 + 7 = -2(1) + 7 = 5 $$

Point: \( (5, 5) \)

For \( x = 2 \):

$$ g(2) = -2(2 - 4)^2 + 7 = -2(-2)^2 + 7 = -2(4) + 7 = -1 $$

Point: \( (2, -1) \)

For \( x = 6 \):

$$ g(6) = -2(6 - 4)^2 + 7 = -2(2)^2 + 7 = -2(4) + 7 = -1 $$

Point: \( (6, -1) \)

Answer:

To graph the parabola, plot the following key points on the coordinate plane:

• Vertex: \( (4, 7) \)
• Other points: \( (3, 5) \), \( (5, 5) \), \( (2, -1) \), and \( (6, -1) \)

Since \( a = -2 \) is negative, the parabola opens downward.