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Difficulty: EasyQuadratic Functions and Graphs

A parabola in the xyxy-plane has its vertex at (3,8)(3, 8) and passes through the point (5,16)(5, 16). If the equation of the parabola is written in the form y=a(x−3)2+8y = a(x - 3)^2 + 8, what is the value of the constant aa?

Answer: 2

Answer

2
Substituting the coordinates (5,16)(5, 16) into y=a(x−3)2+8y = a(x - 3)^2 + 8 yields 16=a(5−3)2+816 = a(5 - 3)^2 + 8. Simplifying the term (5−3)2(5 - 3)^2 gives 44, leading to the equation 16=4a+816 = 4a + 8. Subtracting 8 from both sides gives 8=4a8 = 4a, and dividing by 4 gives the final answer of 2.

Step-by-Step Solution

1
Substitute the point (5,16)(5, 16) into the equation.
16=a(5−3)2+816 = a(5 - 3)^2 + 8
Since the parabola passes through the point (5,16)(5, 16), these coordinates must satisfy the equation.
2
Simplify the squared term.
16=4a+816 = 4a + 8
Subtract 3 from 5 to get 2, and square 2 to get 4.
3
Solve for the constant aa.
a=2a = 2
Subtract 8 from both sides to get 8=4a8 = 4a, then divide by 4 to find aa.

Key Concept

Determining a constant in a quadratic function given its vertex form and a point on the graph.
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Topics

Quadratic Functions and Graphs

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