Answer:
f(x) = - (x - 3) (x + 4)
Explanation:
* Lets explain the graph
- The graph is a parabola, which is the graph of the quadratic function
- The general form of the quadratic function is f(x) = ax² + bx + c
- a is the coefficient of x², if a > 0 the parabola is oped upward, if a < 0
the parabola is opened down ward
- c is the y-intercept of the parabola means the curve intersect the
y-axis at point (0 , c)
- The roots (zeroes) of the quadratic function are the x-intercept of the
parabola means values of x when f(x) = 0
* Now lets solve the problem
- The parabola is downward
∴ The coefficient of x² is negative
- The y-intercept is 12
∴ c = 12
- The x-intercepts are 3 , -4
∴ The zeroes of the function are 3 , -4
∴ x = 3 ⇒ subtract 3 from both sides
∴ x - 3 = 0
∴ x = -4 ⇒ add 4 for both sides
∴ x + 4 = 0
- The factors of f(x) are (x - 3) and (x + 4)
∴ f(x) = -(x - 3)(x + 4)
- Lets find the general form of the function to be sure from the answer
- Multiply the two brackets
∵ f(x) = - [(x)(x) + (x)(4) + (-3)(x) + (-3)(4)] = - [x² + 4x + -3x + -12]
∴ f(x) = - [x² + x - 12] ⇒ multiply the bract by the (-)
∴ f(x) = -x² - x + 12
- Lets check the value of the y-intercept
∵ a = -1 , c = 12
∴ The coefficient of x² is -ve ⇒ the parabola is downward
∴ The y-intercept is 12
∴ f(x) = - (x - 3) (x + 4) is the answer