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.