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The graph of a quadratic function f(x) passes through (9, 0), (0, 9), and (7, 30). What is the equation of axis of symmetry of f(x)?
A. x = -17/4
B. y = -17/4
C. x = 1/4
D. x = 17/4
E. y = 17/4
Are You Up For the Challenge: 700 Level Questions
A. x = -17/4
B. y = -17/4
C. x = 1/4
D. x = 17/4
E. y = 17/4
Are You Up For the Challenge: 700 Level Questions
13 Apr 2022, 07:40
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The standard form of a quadratic is: \(y = ax^2 + bx + c\). We need the values of \(a\) and \(b\) to solve the axis of symmetry, the formula for which is \(x = -\frac{b}{2a}\).
As we are not given a equation, we are dealing with 3 unknowns (a, b and c) and thus need 3 equations to solve for the unknowns. To solve for these we plug in the x and y coordinates of each point into the formula
(0, 9): As this is the y-intercept, we know that 9 will be the value of c. To double check:
\(9 = a(0)^2 + b(0) + c\)
\(9 = a(0)^2 + b(0) + c\)
\( c = 9\)
We can now plug in the value of c for the other two points.
(9, 0): \(0 = a(9)^2 + b(9) + 9\)
\(-9 = 81a + 9b\)
\(-1 = 9a + b\) let this be equation (1)
(7, 30): \(30 = a(7)^2 + b(7) + 9\)
\(21 = 49a + 7b\)
\(3 = 7a + b\) let this be equation (2)
Solving for a and b:
Subtract (2) from (1): \(2a = -4\), which means that \(b = 17\).
The quadratic equation here is: \(y = -2x^2 + 17x + 9\)
Axis of symmetry: \(x = -\frac{b}{2a}\)
\(x = -\frac{17}{(-4)}\)
\(x = \frac{17}{4}\)
Answer D
As we are not given a equation, we are dealing with 3 unknowns (a, b and c) and thus need 3 equations to solve for the unknowns. To solve for these we plug in the x and y coordinates of each point into the formula
(0, 9): As this is the y-intercept, we know that 9 will be the value of c. To double check:
\(9 = a(0)^2 + b(0) + c\)
\(9 = a(0)^2 + b(0) + c\)
\( c = 9\)
We can now plug in the value of c for the other two points.
(9, 0): \(0 = a(9)^2 + b(9) + 9\)
\(-9 = 81a + 9b\)
\(-1 = 9a + b\) let this be equation (1)
(7, 30): \(30 = a(7)^2 + b(7) + 9\)
\(21 = 49a + 7b\)
\(3 = 7a + b\) let this be equation (2)
Solving for a and b:
Subtract (2) from (1): \(2a = -4\), which means that \(b = 17\).
The quadratic equation here is: \(y = -2x^2 + 17x + 9\)
Axis of symmetry: \(x = -\frac{b}{2a}\)
\(x = -\frac{17}{(-4)}\)
\(x = \frac{17}{4}\)
Answer D
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