Answer:
The values of a and c that make the vertex of f(x) equal to (2, -6) are a = 1 and c = 2.
Explanation:
To find the values of a and c such that the vertex of f(x) is (2, -6), we can use the vertex form of a quadratic equation.
The vertex form of a quadratic equation is given by f(x) = a(x - h)² + k, where (h, k) represents the coordinates of the vertex.
In this case, we are given that the vertex is (2, -6). Therefore, we have h = 2 and k = -6.
Now, let's substitute these values into the vertex form equation and simplify it to find the values of a and c.
f(x) = a(x - 2)² - 6
We can expand the equation using the distributive property:
f(x) = a(x² - 4x + 4) - 6
Next, we can simplify further:
f(x) = ax² - 4ax + 4a - 6
Now, we can compare this equation to the given equation f(x) = ax² - 4x - c to find the values of a and c:
Comparing the coefficients, we have:
- The coefficient of x² is a, so a = a.
- The coefficient of x is -4a, so -4a = -4.
- The constant term is 4a - 6, so -c = 4a - 6.
From the second equation, we can solve for a:
-4a = -4
a = 1
Substituting this value of a into the third equation, we can solve for c:
-c = 4a - 6
-c = 4(1) - 6
-c = 4 - 6
-c = -2
c = 2
Therefore, the values of a and c that satisfy the given conditions are a = 1 and c = 2.
To summarize:
- The values of a and c that make the vertex of f(x) equal to (2, -6) are a = 1 and c = 2.