Chapter 9.1, Problem 76PFA

To determine

To evaluate which statement best describes the given graph

Answer to Problem 76PFA

D. The range is $\left\{y\right|y\ge -3\}$ best describes the given graph.

Explanation of Solution

Given information :

Options.

A. The equation of the axis of symmetry is $x=-3$

B. The y-intercept is 1

C. The maximum value is 6

D. The range is $\left\{y\right|y\ge -3\}$

The graph of a quadratic equation ( $y=a{x}^2+bx+c$ ) is a parabola. The parabola is symmetric in nature and the point of symmetry is known as the vertex. This vertex lies on the line of symmetry and is symmetric about this line.

This vertex point shall be:

Highest point (if $a<0$ ) and be called the maximum. Here, the graph opens downwards.

Or, lowest point (if $a>0$ ) and can be called the minimum. Here the graph opens upwards.

In this case, ‘a’ is lesser than 0 hence the graph will have a maximum and will open downwards.

A parabola always points to infinity, either negative or positive.

To graph a quadratic function, compute the axis of symmetry, vertex and y-intercept, post which, plot the same on a graph.

The axis of symmetry bisects the parabola into two equal parts. Hence each point on the parabola would have an equal point on the other side of the axis of symmetry. Plot these points on the graph with a smooth curve.

A. The equation of the axis of symmetry is $x=-3$

The graph shows the vertex at $(3,-3)$ , hence the axis of symmetry is $x=3$

Therefore, option A is false.

B. The y-intercept is 1

The graph shows that the y-axis is intercepted above $(0,4)$

Therefore, option B is false.

C. The maximum value is 6

The parabola in the graph opens upwards and hence it only has a minimum value corresponding to the vertex at the lowest point in the graph. There is no maximum value.

Therefore, option C is false.

D. The range is $\left\{y\right|y\ge -3\}$

The graph shows the parabola with the vertex at the lowest point at $(3,-3)$ . The above statement is correct.

Therefore, option D is true.