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

[Vincen]

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

Answer: B

Source: GMAT Prep

[Brent@GMATPrepNow]

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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

Answer: B

Source: GMAT Prep

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 = 0
Plug x = 0 into the FIRST equation to get: y = (0 + 1)(0 - 1)² = 1
So, 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 = 3
So, 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 = 1
Plug x = 1 into the FIRST equation to get: y = (1 + 1)(1 - 1)² = 0
So, 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 = 2
So, 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