Video: SAT Practice Test 1 • Section 3 • Question 7

The graph in the 𝑥𝑦-plane of the function 𝑦 = 𝑓(𝑥) is shown. If the graph of 𝑦 = −7𝑥 + 2 were drawn on the same axes, which of the following would not be a point of intersection of the two graphs? [A] (0, 0) [B] (1, −5) [C] (−2, 16) [D] (−1, 9)

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Video Transcript

The graph in the 𝑥𝑦-plane of the function 𝑦 equals 𝑓 of 𝑥 is shown. If the graph of 𝑦 equals negative seven 𝑥 plus two were drawn on the same axes, which of the following would not be a point of intersection of the two graphs? Is it A) zero, zero; B) one, negative five; C) negative two, 16; or D) negative one, nine?

The equation 𝑦 equals negative seven 𝑥 plus two is written in the form 𝑦 equals 𝑚𝑥 plus 𝑏. This means that it will have a slope equivalent to 𝑚 and a 𝑦-intercept equivalent to 𝑏. The straight-line graph 𝑦 equals negative seven 𝑥 plus two has a slope of negative seven and a 𝑦-intercept of two or positive two.

As the 𝑦-intercept is equal to positive two, the graph will pass through the point or coordinate zero, two. As the slope of the graph is negative seven, for every one unit we move in the 𝑥-direction, we need to move negative seven in the 𝑦-direction. This means that the graph will also pass through the point one, negative five, as two minus seven is equal to negative five. We can draw a straight line through these points as shown on the graph.

As our graph is quite small, it is not clear exactly which points the graph passes through. We will now go through a method to show that it passes through negative one, nine and negative two, 16. If we consider the 𝑥-coordinates negative two, negative one, zero, and one, we can work out their corresponding 𝑦-values by substituting the 𝑥-values into the equation 𝑦 equals negative seven 𝑥 plus two.

Substituting in 𝑥 equals negative two gives us 𝑦 is equal to negative seven multiplied by negative two plus two. Negative seven multiplied by negative two is equal to 14, as multiplying two negative numbers gives us a positive. Adding two to this gives us an answer of 16. This means that our first coordinate or point that lies on the straight line 𝑦 equals negative seven 𝑥 plus two is negative two, 16.

Substituting in 𝑥 equals negative one gives us 𝑦 is equal to negative seven multiplied by negative one plus two. Negative seven multiplied by negative one is equal to seven. Adding two to this gives us an answer of 𝑦 equals nine. This means that our second coordinate that lies on the graph 𝑦 equals negative seven 𝑥 plus two is negative one, nine. We can repeat this process with 𝑥 equals zero, which gives us a value of 𝑦 equal to two. Therefore, the third coordinate that lies on the line is zero, two.

Finally, substituting 𝑥 equals one into the equation 𝑦 equals negative seven 𝑥 plus two gives us a 𝑦-value equal to negative five. Our fourth coordinate is one, negative five.

We were asked in the question to work out which of the four points given would not be a point of intersection of the two graphs. This means that it would not lie on the function 𝑓 of 𝑥 and on the graph 𝑦 equals negative seven 𝑥 plus two. We have shown from our table and the graph that the three points — negative two, 16; negative one, nine; and one, negative five — all lie on the graph 𝑦 equals negative seven 𝑥 plus two. As the point zero, two lies on the graph 𝑦 equals negative seven 𝑥 plus two, then the point zero, zero is not a point of intersection of the two graphs.

Whilst the function 𝑦 equals 𝑓 of 𝑥 passes through the point zero, zero, it is clear from our table and the straight line on our graph that the line 𝑦 equals negative seven 𝑥 plus two does not pass through this point.

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