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Part 1

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

f left parenthesis x right parenthesis equals 4 x squared plus 16 x minus 20f(x)=4x2+16x−20

We are given the quadratic function:

\[
f(x) = 4x^2 + 16x - 20
\]

Let’s break this down step-by-step for each part:

### Part (a): Finding the Vertex
The vertex of a parabola in the form \( f(x) = ax^2 + bx + c \) can be found using the formula for the x-coordinate of the vertex:

\[
x = \frac{-b}{2a}
\]

For \( f(x) = 4x^2 + 16x - 20 \), we have:
- \( a = 4 \)
- \( b = 16 \)

Substituting into the formula:

\[
x = \frac{-16}{2(4)} = \frac{-16}{8} = -2
\]

Now, to find the y-coordinate of the vertex, substitute \( x = -2 \) back into the function \( f(x) \):

\[
f(-2) = 4(-2)^2 + 16(-2) - 20
\]
\[
f(-2) = 4(4) + (-32) - 20 = 16 - 32 - 20 = -36
\]

Thus, the vertex is at \( (-2, -36) \).

### Part (b): Maximum or Minimum Value
Since the coefficient of \( x^2 \) (which is \( a = 4 \)) is positive, the parabola opens upwards. This means the vertex is a **minimum** point.

The minimum value of the function is the y-coordinate of the vertex, which is \( -36 \).

### Part (c): Finding the Range
Since the parabola opens upwards and the vertex is the minimum point, the range of the function is:

\[
\text{Range} = [-36, \infty)
\]

### Part (d): Increasing and Decreasing Intervals
For a parabola \( f(x) = ax^2 + bx + c \), the function:
- Decreases to the left of the vertex
- Increases to the right of the vertex

Since the vertex is at \( x = -2 \):
- The function is **decreasing** on \( (-\infty, -2) \)
- The function is **increasing** on \( (-2, \infty) \)

### Summary
- **Vertex**: \( (-2, -36) \)
- **Minimum value**: \( -36 \)
- **Range**: \( [-36, \infty) \)
- **Increasing interval**: \( (-2, \infty) \)
- **Decreasing interval**: \( (-\infty, -2) \)

Do you want more details or have any questions about these steps?

Here are five related questions you can explore:
1. How would the vertex change if the coefficient of \( x^2 \) were negative?
2. What are the domain and axis of symmetry for this function?
3. How does completing the square help in finding the vertex?
4. How do we find the roots of this quadratic function?
5. How can we determine the direction of a parabola without graphing it?

**Tip**: The vertex formula \( x = \frac{-b}{2a} \) is a quick way to find the turning point of any quadratic function.

Given the following function, (a) find the vertex; (b) determine whether there is a maximum or a minimum value, and find the value; (c) find the range; and (d) find the intervals on which the function is increasing and the intervals on which the function is decreasing. f(x) = 4x^2 + 16x - 20

Explore how to solve the quadratic function f(x) = 4x^2 + 16x - 20. This detailed solution guides you through finding the vertex, determining minimum or maximum value, calculating the range, and identifying increasing and decreasing intervals. Ideal for students in Grades 9-11.

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