27-f(x)=1/4x^2-2x-12. Can you show me how to find the vertex and x intercepts, step-by-step? I tried to put it in standard form, with (1/2x-1)^2-13, but my book said that's not the answer. Thanks!
Is 27 the problem number or is that part of the equation?
If 27 is part of the equation, then
f(x) = -(1/4)x^2 + 2x +39. That is not anything like your equation.
If the 27 is a problem number, you should not have types it as part of the equation. Is the minus sign in front of f(x) real, or a hyphen?
We can only answer math questions that are clearly written.
It's just the number of the problem.
Of course! I'll walk you through the process step-by-step.
To find the vertex and x-intercepts of the given quadratic equation, we need to rewrite it in the standard form of a quadratic equation, which is in the form of "f(x) = ax^2 + bx + c".
Let's start by simplifying the equation you provided:
27 - f(x) = 1/4x^2 - 2x - 12
To get rid of the negative sign in front of f(x), let's multiply both sides of the equation by -1:
f(x) - 27 = -(1/4)x^2 + 2x + 12
Now, let's rearrange the terms to get the equation in the standard form:
f(x) = -(1/4)x^2 + 2x + 12 + 27
f(x) = -(1/4)x^2 + 2x + 39
Now that we have the equation in standard form, we can easily identify the values of a, b, and c:
a = -(1/4)
b = 2
c = 39
The x-coordinate of the vertex can be found using the formula: x = -b/2a. Let's substitute the values of a and b into the formula:
x = -2 / (2 * -(1/4))
x = -2 / (-1/2)
x = -2 * (-2/1)
x = 4
Now, to find the y-coordinate of the vertex, we substitute the x-coordinate we just found (x = 4) into the original equation:
f(4) = -(1/4) * (4)^2 + 2 * (4) + 39
f(4) = -1 * 16/4 + 8 + 39
f(4) = -4 + 8 + 39
f(4) = 43
Therefore, the vertex of the quadratic equation is (4, 43).
To find the x-intercepts, we set f(x) = 0 and solve for x. In other words, we find the values of x when the parabola intersects the x-axis:
-(1/4)x^2 + 2x + 39 = 0
We can solve this equation by factoring, completing the square, or using the quadratic formula. However, in this case, factoring is not straightforward. So, let's use the quadratic formula:
The quadratic formula states that x = (-b ± √(b^2 - 4ac)) / (2a)
By substituting the values from our equation, we get:
x = (-2 ± √((2)^2 - 4 * -(1/4) * 39)) / (2 * -(1/4))
x = (-2 ± √(4 + 39))/(-1/2)
x = (-2 ± √43) / (-1/2)
To simplify this expression further, we multiply both the numerator and denominator by -2:
x = (2 ± √43) / (1/2)
x = (2 ± √43) * (2/1)
x = 4 ± 2√43
So, the x-intercepts, also known as the roots or solutions, of the quadratic equation are x = 4 + 2√43 and x = 4 - 2√43.
That's it! You have successfully found the vertex and x-intercepts of the quadratic equation. Remember, the standard form is essential when applying formulas and finding key points on the graph.