Video: Graphing Functions

Which of the following is the function represented by the shown graph? [A] 𝑦 = (3/2)π‘₯ + 2 [B] 𝑦 = 2π‘₯ + (2/3) [C] 𝑦 = (2/3)π‘₯ βˆ’ 2 [D] π‘₯ = (2/3)𝑦 + 2 [E] 𝑦 = (2/3)π‘₯ + 2

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Video Transcript

Which of the following is the function represented by the shown graph? Option A, 𝑦 equals three-halves π‘₯ plus two. Option B, 𝑦 equals two π‘₯ plus two-thirds. Option C, 𝑦 equals two-thirds π‘₯ minus two. Option D, π‘₯ equals two-thirds 𝑦 plus two. Or option E, 𝑦 equals two-thirds π‘₯ plus two.

Let’s recall that the general form of a linear equation is 𝑦 equals π‘šπ‘₯ plus 𝑏 or 𝑦 equals π‘šπ‘₯ plus 𝑐, where the constant term of 𝑏 or 𝑐 represents the 𝑦-intercept of the function. The value of π‘š will indicate the slope or gradient of the line. So therefore, if we were to calculate the slope and the 𝑦-intercept of this drawn function, we could work out what the function is. We can recall that, between two coordinates π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two, the slope is equal to 𝑦 two minus 𝑦 one over π‘₯ two minus π‘₯ one.

We can select any two coordinates on the line for π‘₯ one, 𝑦 one and π‘₯ two, 𝑦 two. But often the easiest ones to pick are those which have integer values. We can see here that zero, two and three, four both lie on the line. It doesn’t matter which coordinate we designate as π‘₯ one, 𝑦 one and which we designate as π‘₯ two, 𝑦 two. To find the slope, we substitute in our 𝑦 two and 𝑦 one values to give us four minus two over the π‘₯ two minus π‘₯ one values, which is three minus zero. Simplifying this, we have a slope of two-thirds.

To find the 𝑦-intercept, we look at the graph to see where it crosses the 𝑦-axis. And that happens when the 𝑦-value is two. We can then fill in our values of slope and 𝑦-intercept into the general equation. We found that the slope π‘š is equal to two-thirds and the 𝑦-intercept of 𝑏 is equal to two. And so our answer is that given in option E, 𝑦 equals two-thirds π‘₯ plus two.

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