bySayantani Barman Experta en el extranjero
Question:
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?
- None
- One
- Two
- Three
- Four
“The figure shows the graph of y = (x + 1)(x - 1)^2 in the xy-plane." - is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been borrowed from the book “GMAT Official Guide Quantitative Review”.
To understand GMAT Problem Solving questions, applicants must possess fundamental qualitative skills. Quant tests a candidate's aptitude in reasoning and mathematics. The GMAT Quantitative test's problem-solving phase consists of a question and a list of possible responses. By using mathematics to answer the question, the candidate must select the appropriate response. The problem-solving section of the GMAT Quant topic is made up of very complicated math problems that must be solved by using the right math facts.
Answer: B
Solution and Explanation:
Approach Solution 1:
The graph of f(x) +2 is shifted upwards by two units compared to f(x)
Hence the red curve clearly indicates that the curve will cut the x axis at only one point.
Correct Answer: B
Approach Solution 2:
Let's assume that f(x) equals the value given in this question (f(x) = (x+1)(x-1)^2) and that y = f(x).
Case 1: f(x) → f(x) + a
Move the f(x) graph by a points up the Y-axis while maintaining its shape.
Case 2: f(x) → f(x) - a
Move the f(x) graph by a points southward on the Y-axis while maintaining its shape.
Case 3: f(x) → f(x+a)
Maintaining the same shape, move the f(x) graph by a points to the left on the X-axis.
Case 4: f(x) → f(x-a)
Maintaining the same shape, move the f(x) graph one point to the right on the X-axis.
(Yes, the graph does go in the other way for cases 3 and 4. When f(x+a) is present, it goes to the left (negative X-axis), whereas f(x-a) moves to the right (positive X-axis).
Case 5: f(x) → -f(x)
Consider the mirror image of f (x) on the Y-axis.
Case 6: f(x) → f(-x)
Turn the f(x) graph 180 degrees from the X-axis.
In light of all the available information, we can thus conclude that the curve will only cross the x axis once. The correct response is B.
Correct Answer: B
Approach Solution 3:
The problem actually asks for the number of intercepts, not the exact intercepting point.
So let's consider (x+1)(x-1)^2 +2=0 and
change the equation into (x+1)(x-1)^2 = -2.
Then the problem is changed to find the number of intercepts between y = (x+1)(x-1)^2 and y=-2.
Looking at the graph, you would easily find that y=(x+1)(x-1)^2 intercepts y=-2
(which is the line parallel to the x-axis passing (0,-2)) only once. So the answer is B.
Correct Answer: B
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