Chapter 9.3, Problem 38PPS

a.

To determine

To graph the given function.

Explanation of Solution

Given information :

Given function is $y=x^2-2x-3$ .

Graph :

Interpretation :

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 more than 0 hence the graph will have a minimum and will open upwards.

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.

Formula to compute equation of the axis of symmetry $x=-\frac{b}{2\times a}$ . Here, $a=-3$ and $b=6$ . ‘a’ and ‘b’ are coefficients $x^2$ and $x$ respectively.

Axis of symmetry for the given function is

  $x=-\frac{b}{2\times a}$

Formula for axis of symmetry.

  $x=-\frac{-2}{2\times \left(1\right)}$

Putting the values of ‘a’ and ‘b’ .

  $=\frac{2}{2}$

Simplifying this

  $x=1$ is the equation of axis of symmetry.

Vertex can be found out by putting the value of x computed in the axis of symmetry in the original function. This will give a value of y . These two coordinates of x and y would be the point where the vertex is.

  $={\left(1\right)}^2-2\left(1\right)-3$

Putting the value of $x=1$ in the original function.

  $=1-2-3$

Simplifying the expression.

  $=-4$

Thus the vertex is $\left(1,-4\right)$ .

y-intercept is computed by substituting the value of x in the equation by 0.

  $={\left(0\right)}^2-2\left(0\right)-3$

Simplifying

  $=-3$

Hence the point of y-intercept is $\left(0,-3\right)$ .

Now, plot these points along with their reflecting symmetric points, starting from the vertex.

b.

To determine

To name the zeros of the function.

Answer to Problem 38PPS

The zeros of the function are $\left(-1,0\right)$ and $\left(3,0\right)$

Explanation of Solution

Given information :

The function is $y=x^2-2x-3$ .

Using the graphing calculator, the graph for the given function is:

Zeros of the parabola are the places where the parabola intersects the x-axis.

This given graph, the zeros are at $\left(-1,0\right)$ and $\left(3,0\right)$

c.

To determine

To calculate factors of the given function.

Answer to Problem 38PPS

The factors are $\left(x+1\right)$ and $\left(x-3\right)$ .

Explanation of Solution

Given information :

The function provided is $x^2-2x-3=0$ .

Formula used :

Factoring method involves in factorizing the terms in order to arrive at values that, if multiplied with each other, would provide the original quadratic function.

Calculation :

The graph of the given equation is $x^2-2x-3=0$

  $=>x^2-3x+x-3=0$

Breaking the middle term that when multiplied, provides that values of $a{x}^2\times c$ .

  $=>x\left(x-3\right)+1\left(x-3\right)=0$

Arranging the common elements.

  $=>\left(x-3\right)\left(x+1\right)=0$

The factors are $\left(x+1\right)$ and $\left(x-3\right)$ .

d.

To determine

To equate each factor with zero and find the result.

Answer to Problem 38PPS

The results of x after equating with zero are -1 and 3 .

Explanation of Solution

Given information :

The function provided is $x^2-2x-3=0$ .

The graph of the given equation is $x^2-2x-3=0$

The factors are $\left(x+1\right)$ and $\left(x-3\right)$ .

  $x-3=0$

Equating the factors with zero.

  $x=3$

This is one result.

  $x+1=0$

Equating the other factor with zero.

  $x=-1$

This is the other result.

e.

To determine

To draw conclusion regarding the factors of a quadratic equation.

Answer to Problem 38PPS

The factors of a quadratic equation, if simplified with zero, would give its roots.

Explanation of Solution

Given information :

The function provided is $x^2-2x-3=0$ .

There are two factors of a quadratic equation. These factors, if multiplied, would yield the main equation. Also, if they are equated with zero, they provide the roots of the same quadratic equation.

When graphed, these roots are also called the zeros. They are the points where the graph meets the x-axis.