Given the following function, (a) find the vertex, (b) determine whether there is a maximum or a minimum value, and find the value; © find the range; and (d) find the intervals on which the function
is increasing and the intervals on which the function is decreasing

(a) The vertex is
(Type an ordered pair, usiipg integers or fractions.)

Ask by Medina Hart. in the United States

Nov 07,2024

Question

Given the following function, (a) find the vertex, (b) determine whether there is a maximum or a minimum value, and find the value; © find the range; and (d) find the intervals on which the function
is increasing and the intervals on which the function is decreasing

(a) The vertex is
(Type an ordered pair, usiipg integers or fractions.)

Ask by Medina Hart. in the United States

Nov 07,2024

UpStudy AI · Accepted Answer

To analyze the function $$ f(x) = -\frac{1}{2}x^2 + 7x - 2 $$, we can follow the steps outlined in your question.

### (a) Find the vertex

The vertex of a quadratic function in the form $$ f(x) = ax^2 + bx + c $$ can be found using the formula for the x-coordinate of the vertex:

$$
x = -\frac{b}{2a}
$$

Here, $$ a = -\frac{1}{2} $$ and $$ b = 7 $$.

Calculating the x-coordinate of the vertex:

$$
x = -\frac{7}{2 \cdot -\frac{1}{2}} = -\frac{7}{-1} = 7
$$

Now, we substitute $$ x = 7 $$ back into the function to find the y-coordinate:

$$
f(7) = -\frac{1}{2}(7^2) + 7(7) - 2
$$
$$
= -\frac{1}{2}(49) + 49 - 2
$$
$$
= -24.5 + 49 - 2
$$
$$
= 22.5
$$

Thus, the vertex is:

$$
\text{Vertex} = (7, 22.5)
$$

### (b) Determine whether there is a maximum or a minimum value, and find the value

Since the coefficient of $$ x^2 $$ (which is $$ a = -\frac{1}{2} $$) is negative, the parabola opens downwards. This means that the vertex represents a maximum value.

The maximum value of the function is:

$$
\text{Maximum value} = 22.5
$$

### (c) Find the range

Since the parabola opens downwards and the maximum value is 22.5, the range of the function is:

$$
\text{Range} = (-\infty, 22.5]
$$

### (d) Find the intervals on which the function is increasing and the intervals on which the function is decreasing

The function is increasing on the interval to the left of the vertex and decreasing on the interval to the right of the vertex.

- The function is increasing on $$ (-\infty, 7) $$.
- The function is decreasing on $$ (7, \infty) $$.

### Summary of Results

(a) The vertex is $$ (7, 22.5) $$.

(b) There is a maximum value of $$ 22.5 $$.

(c) The range is $$ (-\infty, 22.5] $$.

(d) The function is increasing on $$ (-\infty, 7) $$ and decreasing on $$ (7, \infty) $$.
(a) The vertex is $$ (7, 22.5) $$. (b) The function has a maximum value of $$ 22.5 $$. (c) The range is $$ (-\infty, 22.5] $$. (d) The function is increasing on $$ (-\infty, 7) $$ and decreasing on $$ (7, \infty) $$.

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