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Given f(x)=ax²-4x-c, find the values of a and c such that the vertex of f(x) is (2,-6).

User Jluckyiv
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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.

User Yaroslav
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