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Math  /  Algebra

QuestionThe graph of a quadratic function passes through A(3,12)\mathrm{A}(3,12) and has xx-intercepts 1 and 5 . Write an equation of the graph in factored form.

Studdy Solution

STEP 1

1. The quadratic function can be expressed in factored form as f(x)=a(x−r1)(x−r2) f(x) = a(x - r_1)(x - r_2) , where r1 r_1 and r2 r_2 are the roots (x-intercepts).
2. The given x-intercepts are x=1 x = 1 and x=5 x = 5 .
3. The function passes through the point A(3,12) \mathrm{A}(3,12) , which can be used to solve for the coefficient a a .

STEP 2

1. Write the general form of the quadratic function using the given x-intercepts.
2. Substitute the coordinates of point A(3,12) \mathrm{A}(3,12) into the equation to solve for a a .
3. Write the final equation of the quadratic function in factored form.

STEP 3

Write the general form of the quadratic function using the given x-intercepts. Since the x-intercepts are 1 1 and 5 5 , the function can be written as:
f(x)=a(x−1)(x−5) f(x) = a(x - 1)(x - 5)

STEP 4

Substitute the coordinates of point A(3,12) \mathrm{A}(3,12) into the equation to solve for a a . The point A(3,12) \mathrm{A}(3,12) means that when x=3 x = 3 , f(x)=12 f(x) = 12 . Substitute these values into the equation:
12=a(3−1)(3−5) 12 = a(3 - 1)(3 - 5)

STEP 5

Simplify the equation to solve for a a :
12=a(2)(−2) 12 = a(2)(-2) 12=−4a 12 = -4a

STEP 6

Solve for a a by dividing both sides by −4-4:
a=12−4 a = \frac{12}{-4} a=−3 a = -3

STEP 7

Write the final equation of the quadratic function in factored form using the value of a a found in the previous step:
f(x)=−3(x−1)(x−5) f(x) = -3(x - 1)(x - 5)
The equation of the graph in factored form is:
−3(x−1)(x−5) \boxed{-3(x - 1)(x - 5)}

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