Chapter 3, Problem 1RE

To determine

a) The sign of the leading coefficient

b) Vertex

c) Axis of symmetry

d) Intervals where f is increasing and where f is decreasing

Answer to Problem 1RE

Solution:

1. The sign of the leading coefficient must be negative.
2. The vertex is $(2,4)$ .
3. $x=2$ is the axis of symmetry.
4. f is increasing on $\left(-\infty ,2\right)$ and decreasing on $\left(2,\infty \right)$ .

Explanation of Solution

Given:

The graph of the parabola

Calculation:

1. We know that for an upward parabola the leading coefficient is positive and for a downward parabola the leading coefficient is negative. From the given graph, we can see that the parabola is downward. Thus, the sign of the leading coefficient must be negative.
2. For a downward parabola, vertex is the maximum point on the parabola. The maximum point on the given parabola is $(2,4)$ . Therefore, the vertex is $(2,4)$ .
3. Axis of symmetry divides a parabola into two equal halves and passes through the vertex and the focus. From the given graph we can see that the line $x=2$ divides the parabola in two equal halves and, also passes through the vertex $(2,4)$ . Therefore, $x=2$ is the axis of symmetry.
4. From the given graph, f is increasing on $\left(-\infty ,2\right)$ and decreasing on $\left(2,\infty \right)$ .

Conclusion:

Using the concept of parabola, we have found that:

1. The sign of the leading coefficient must be negative.
2. The vertex is $(2,4)$ .
3. $x=2$ is the axis of symmetry.
4. f is increasing on $\left(-\infty ,2\right)$ and decreasing on $\left(2,\infty \right)$ .