Video Transcript
Which of the following graphs represents the equation 𝑦 equals negative five 𝑥 minus two?
And then we have five graphs: (A), (B), (C), (D), and (E). To answer the question, we’ll think about the line represented by the equation 𝑦 equals negative five 𝑥 minus two. The equation of this line has been given to us in the slope–intercept form: 𝑦 equals 𝑚𝑥 plus 𝑏, where 𝑚, the coefficient of 𝑥, represents the slope of the line and 𝑏, the constant term, represents its 𝑦-intercept.
By comparing the equation of our line with the slope–intercept form, we can determine the values of 𝑚 and 𝑏, the slope and 𝑦-intercept for this line, which will reveal some key features about its graph. First, we see the value of the constant term in this equation is negative two. And so the 𝑦-intercept of this line is negative two. This means that the graph intercepts the 𝑦-axis at a value of negative two.
Looking at the five graphs then, we can determine the values at which each of these lines intercept the 𝑦-axis. Graph (A) intercepts the 𝑦-axis at positive two. And so this rules out option (A). Graph (B), however, intersects the 𝑦-axis at negative two. So graph (B) is still a possibility. Graph (C) intercepts the 𝑦-axis at a value of negative five. So this has the incorrect 𝑦-intercept and rules out graph (C). Graphs (D) and (E), however, both intercept the 𝑦-axis at negative two. So each of these are still possible.
Next, we return to the equation of this line and we see that the slope, which is the coefficient of 𝑥 in this equation, is equal to negative five. This means that for every one unit the line moves to the right, it will move down by five units. We can rule out option (B) straightaway because we see that, in this graph, the line slopes upwards from left to right. And so this line has a positive slope.
We’re left with only options (D) and (E). To determine the slope of these lines, we need to consider two points. We’ve already identified one point on each graph. That’s its 𝑦-intercept. So let’s consider a second point. On graph (D), we identify the point with coordinates negative one, three, which lies on this line. We then see that to get from this point to the 𝑦-intercept of the graph, we move one unit to the right and five units down, which confirms that the slope of graph (D) is negative five.
We think our answer is graph (D) then as it has the correct slope and the correct 𝑦-intercept. But let’s just check the slope for graph (E). This time, we identify the point with coordinates negative five, negative one, which lies on this line. To get from this point to the 𝑦-intercept, we go five units to the right and one unit down. Remembering that the slope of a line is its change in 𝑦 over its change in 𝑥, we see that the slope of graph (E) is in fact negative one-fifth. It is the reciprocal of the value we were looking for.
Our answer then is that graph (D) represents the equation 𝑦 equals negative five 𝑥 minus two because it has a slope of negative five and a 𝑦-intercept of negative two.
