Please help I can't figure out these two.
Express f(x) in the form a(x − h)2 + k.
f(x) = −4x2 + 24x − 9
Find the standard equation of a parabola that has a vertical axis and satisfies the given conditions.
vertex (0, −7), passing through (3, 38)
Thank You!
for the first, complete the square:
f(x) = -4(x^2 - 6x ......) - 9
= -4( x^2 - 6x + 9 -9) - 9
= -4( (x-3)^2 - 9) - 9
= -4(x-3)^2 + 36 - 9
= -4(x-3)^2 + 27
for the second, you know the vertex is (0,-7)
so f(x) = a(x-0)^2 - 7
= ax^2 - 7
but (3,38) lies on it, so
38 = a(9)
a = 38/9
f(x) = (38/9)x^2 -7
f(x) = -4(x^2-6x) - 9
= -4(x^2-6x+9) -4(-9) - 9
= -4(x-3)^2 + 27
f(x) = a(x-0)^2 - 7
38 = 9a-7
a = 5
f(x) = 5x^2 - 7
don't know where my - 7 went ????
Thank You Guys!!!
F(x) = -4x^2+24x-9.
h = -B/2A = -24/-8 = 3.
k = -4*3^2 + 24*3 - 9=-36 + 72 - 9 = 27.
F(x) = a(x-h)^2 + k.
F(x) = -4(x-3)^2 + 27.
To express f(x) in the form a(x - h)^2 + k, we need to complete the square.
For the equation f(x) = -4x^2 + 24x - 9:
Step 1: Factor out the coefficient of x^2 from the first two terms: -4(x^2 - 6x) - 9.
Step 2: Take half of the coefficient of x (-6) and square it: (-6/2)^2 = 9.
Step 3: Add and subtract the result obtained in step 2 inside the parentheses: -4(x^2 - 6x + 9 - 9) - 9.
Step 4: Group the terms inside the parentheses: -4[(x - 3)^2 - 9] - 9.
Step 5: Expand the squared term inside the brackets: -4(x - 3)^2 + 36 - 9 - 9.
Step 6: Combine like terms: -4(x - 3)^2 + 18 - 9.
Step 7: Simplify further: -4(x - 3)^2 + 9.
Therefore, f(x) can be expressed in the desired form as f(x) = -4(x - 3)^2 + 9.
Now, let's move on to finding the standard equation of a parabola with a vertical axis that satisfies the given conditions.
The general equation of a parabola with a vertical axis is: y = a(x - h)^2 + k, where (h, k) is the vertex of the parabola.
Given that the vertex is (0, -7) and the parabola passes through the point (3, 38), we can substitute these values into the equation.
Step 1: Substitute the vertex coordinates into the equation: y = a(x - 0)^2 - 7.
Simplified: y = ax^2 - 7.
Step 2: Substitute the coordinates (3, 38) into the equation to solve for 'a': 38 = a(3)^2 - 7.
Simplified: 38 = 9a - 7.
Step 3: Solve for 'a': 9a = 38 + 7.
Simplified: 9a = 45.
Step 4: Divide both sides by 9: a = 45/9.
Simplified: a = 5.
Therefore, the equation of the parabola with a vertical axis, satisfying the given conditions, is y = 5x^2 - 7.