Chapter 9.1, Problem 78PFA

To determine

To identify the correct option that represents the given graph.

Answer to Problem 78PFA

D. vertex: $(1,-2)$ ; axis of symmetry: $x=1$

Explanation of Solution

Given information :

Graph representing a function.

Options.

A. vertex: $(-2,1)$ ; axis of symmetry: $x=1$

B. vertex: $(-2,1)$ ; axis of symmetry: $y=-2$

C. vertex: $(1,-2)$ ; axis of symmetry: $y=-2$

D. vertex: $(1,-2)$ ; axis of symmetry: $x=1$

The graph represented by the function is that of 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.

In the graph shown, it is evident that the curve is symmetric along the line $x=1$ . Therefore, options B and C are rejected.

The vertex of the curve is to the right of the y -axis, that is, the x -coordinate of the vertex has to be positive. Therefore, option A is rejected.

Option D. vertex: $(1,-2)$ ; axis of symmetry: $x=1$ is the correct answer.