imhimanshu wrote:

The figure shows the graph of y = (x + 1)(x - 1)² in the xy-plane. At how many points does the graph of y = (x + 1)(x - 1)² + 2 intercept the x-axis?

A. None

B. One

C. Two

D. Three

E. Four

Let's find some points that lie on each of the curves.

So, for each equation, we'll find a pair of values (an x-value and a y-value) that satisfy each equation.

We'll do so by plugging in some x-values and calculating the corresponding y-values.

Let's start with x =

0Plug x =

0 into the FIRST equation to get: y = (

0 + 1)(

0 - 1)² =

1So, the point (

0,

1) lies ON the curve defined by y = (x + 1)(x - 1)²

Now, plug x =

0 into the SECOND equation to get: y = (

0 + 1)(

0 - 1)² + 2 =

3So, the point (

0,

3) lies ON the curve defined by y = (x + 1)(x - 1)² + 2

Add the point (

0,

3) to our graph to get:

Notice that the point (0, 3) is 2 UNITS directly ABOVE the point (0, 1)---------------------------------------------

Let's try another x-value....

Try x =

1Plug x =

1 into the FIRST equation to get: y = (

1 + 1)(

1 - 1)² =

0So, the point (

1,

0) lies ON the curve defined by y = (x + 1)(x - 1)²

Now, plug x =

1 into the SECOND equation to get: y = (

1 + 1)(

1 - 1)² + 2 =

2So, the point (

1,

2) lies ON the curve defined by y = (x + 1)(x - 1)² + 2

Add the point (

1,

2) to our graph to get:

Notice that the point (1, 2) is 2 UNITS directly ABOVE the point (1, 0)---------------------------------------------

At this point, we should recognize that the graph of y = (x + 1)(x - 1)² + 2 is very similar to the graph of y = (x + 1)(x - 1)²

The only difference is that the graph of y = (x + 1)(x - 1)² + 2 is SHIFTED UP 2 units.

So, to graph the curve y = (x + 1)(x - 1)² + 2, we can just take every point on the curve y = (x + 1)(x - 1)² and move it UP 2 units...

When we connect the points, we see that the graph of y = (x + 1)(x - 1)² + 2 looks something like this.

From our sketch, we can see that the graph of y = (x + 1)(x - 1)² + 2 intercepts the x-axis ONCE

Answer: B

Cheers,

Brent

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