Chapter 7.6, Problem 39WE

(a)

To determine

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

Answer to Problem 39WE

Zero

Explanation of Solution

Given:

The graph is :

  

Calculation:

Since the given graph intersects x -axis at one point , so , it has two real and equal 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 two equal and real roots, so ,

  $b^2-4ac$ = 0

(b)

To determine

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

Answer to Problem 39WE

Positive.

Explanation of Solution

Given:

The graph is :

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

  

Calculation:

In the equation $y=a{x}^2+bx+c$ , 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 positive.

So, c is positive.

(c)

To determine

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

Answer to Problem 39WE

Positive.

Explanation of Solution

Given:

The graph is :

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

  

Calculation:

In the equation $y=a{x}^2+bx+c$ , 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 upward , so , a is positive.

(d)

To determine

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

Answer to Problem 39WE

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 positive . So, 2a is also positive.

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

So, b is positive.