Questions: (a) Use the vertex formula to find the vertex of the following function. (b) Find the intervals where f is increasing and where f is decreasing. f(x)=-3/4 x^2+1/2 x-4

(a) Use the vertex formula to find the vertex of the following function.
(b) Find the intervals where f is increasing and where f is decreasing.
f(x)=-3/4 x^2+1/2 x-4

Transcript text: (a) Use the vertex formula to find the vertex of the following function. (b) Find the intervals where $f$ is increasing and where $f$ is decreasing. \[ f(x)=-\frac{3}{4} x^{2}+\frac{1}{2} x-4 \]

Solution

Solution Steps

Step 1: Finding the Vertex

To find the vertex of the quadratic function $f(x) = \frac{-3}{4}x^2 + \frac{1}{2}x + (-4)$, we use the vertex formula $x = -\frac{b}{2a}$. Substituting $a = \frac{-3}{4}$ and $b = \frac{1}{2}$, we get $x = \frac{1}{3}$. The $y$-coordinate of the vertex is found by substituting $x$ back into the function: $y = \frac{-3}{4}(\frac{1}{3})^2 + \frac{1}{2}(\frac{1}{3}) + (-4) = \frac{-47}{12}$. Therefore, the vertex is $\left(\frac{1}{3}, \frac{-47}{12}\right)$.

Step 2: Determining Intervals of Increase and Decrease

Since $a = \frac{-3}{4}$, the parabola opens downwards, indicating that the function is increasing on the interval $(-∞, \frac{1}{3})$ and decreasing on the interval $(\frac{1}{3}, ∞)$.

Step 3: Determining Maximum or Minimum Value

The vertex represents the maximum value of the function, which is $\frac{-47}{12}$ at $x = \frac{1}{3}$.

Final Answer:

The vertex of the quadratic function is $\left(\frac{1}{3}, \frac{-47}{12}\right)$, with the function increasing on the interval $(-∞, \frac{1}{3})$ and decreasing on the interval $(\frac{1}{3}, ∞)$. The vertex represents the maximum value of the function.

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