Chapter 7.6, Problem 38WE

(a)

To determine

Find whether $b^2-4ac$ is positive, negative or zero by observing the given graph.

Answer to Problem 38WE

Positive.

Explanation of Solution

Given:

The graph is :

  

Calculation:

Since the given graph intersects the x -axis at 2 distinct points , it has 2 real and distinct roots.

  $b^2-4ac$ is the discriminant , which helps in describing the number and type of roots.

  If $b^2-4ac$ is negative, there is no real root.

  If $b^2-4ac$ is zero, the roots are real and equal.

  If $b^2-4ac$ is positive , the roots are real and distinct.

Since the graph has 2 real and distinct roots,

  $b^2-4ac$ is positive.

(b)

To determine

Find whether c is positive, negative or zero by observing the given graph.

Answer to Problem 38WE

Zero.

Explanation of Solution

Given:

The graph is :

A parabola represented by the equation $y=a{x}^2+bx+c$

  

Calculation:

In the equation $y=a{x}^2+bx+c$ , c is the y -intercept , y-coordinate of the point where the graph intersects the y -axis.

From the graph , it is clear that the y -intercept is 0, since it intersects the y -axis at y = 0.

So, c = 0.

(c)

To determine

Find whether a is positive, negative or zero by observing the given graph.

Answer to Problem 38WE

Negative

Explanation of Solution

Given:

The graph is :

A parabola represented by the equation $y=a{x}^2+bx+c$

  

Calculation:

In the equation $y=a{x}^2+bx+c$ , a represents the direction in which the parabola opens.

  If a > 0 , the parabola opens upward.

  If a < 0 , the parabola opens downward.

Since the parabola in the given graph opens downward , so , a is negative.

(d)

To determine

Find whether b is positive, negative or zero by observing the given graph.

Answer to Problem 38WE

Positive.

Explanation of Solution

Given:

The graph is :

A parabola represented by the equation $y=a{x}^2+bx+c$

  

Calculation:

In the equation $y=a{x}^2+bx+c$ , equation of axis of symmetry is $x=-\frac{b}{2a}$ .From part (c) , a is negative . So, 2a is also negative.

Since , from the graph , it is clear that the axis of symmetry lies on the positive x -axis , we have $x=-\frac{b}{2a}>0$ and 2a is negative , so , for this to be true , we must have b > 0.

So, b is positive.