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Which graph represents the function of
f(x)=(4x^(2)-4x-8)/(2x+2)?

Which graph represents the function of $f(x)=\frac{4 x^{2}-4 x-8}{2 x+2} ?$

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Find the vertex of the transformed function

Full solution

Q. Which graph represents the function of

f(x)=\frac{4 x^{2}-4 x-8}{2 x+2} ?

Simplify and Factor: First, simplify the function $f(x)$ by factoring and reducing the expression where possible. $f(x) = \frac{4x^2 - 4x - 8}{2x + 2}$ Factor out the common factor of $4$ in the numerator and $2$ in the denominator. $f(x) = \frac{4(x^2 - x - 2)}{2(x + 1)}$
Divide by Common Factor: Now, divide both the numerator and the denominator by their greatest common divisor, which is $2$ . $\newline$ $f(x) = \frac{2(x^2 - x - 2)}{(x + 1)}$
Factor Quadratic: Next, we need to check if the quadratic in the numerator can be factored further. The quadratic $x^2 - x - 2$ can be factored into $(x - 2)(x + 1)$ . $f(x) = \frac{2(x - 2)(x + 1)}{(x + 1)}$
Cancel Common Factor: Since $(x + 1)$ is a common factor in both the numerator and the denominator, we can cancel it out, but we must remember that $x = -1$ is a point of discontinuity (a hole in the graph) because we cannot divide by zero. $\newline$ $f(x) = 2(x - 2)$ , for $x \neq -1$
Final Linear Function: The simplified function $f(x) = 2(x - 2)$ is a linear function with a slope of $2$ and a y-intercept at $(0, -4)$ . The graph of this function is a straight line, and we must remember to indicate the point of discontinuity at $x = -1$ .

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