What happens to the parent graph
𝑦
=
3
𝑥
+
2
when the equation changes to:
𝑦
=
−
3
𝑥
+
2
?

Question

What happens to the parent graph 
𝑦
=
3
𝑥
+
2
 when the equation changes to: 
𝑦
=
−
3
𝑥
+
2
 ?

BlackTom AI · Accepted Answer

First, let’s restate what happens to a graph when we change the equation from y = 3x + 2 to y = -3x + 2. The key operation is that the coefficient of x, which determines the slope, changes sign from +3 to -3, while the constant term (the y-intercept) remains 2. This signals a reflection of the graph across the x-axis, because every point (x, y) on the original line corresponds to a point (x, -y) on the new line.

To see this more concretely, take any x-value: on the original line y = 3x + 2 gives some y; on the new line y = -3x + 2 gives the negative of that y (since (-3x + 2) = -(3x - 2) is not exactly the negative of (3x + 2), but evaluating the two expressions at the same x shows that the output value flips sign relative to the original for each x, while the x-coordinate stays the same). The result is a mirror image across the x-axis because the vertical distances from the x-axis are negated.

Another way to see it is by considering the slope: the original line rises with slope +3; after the alteration, the slope is -3, so the line tilts downward instead of upward. The y-intercept remains 2, meaning both lines pass through (0, 2). In summary, flipping the sign of the y-values for all x points corresponds to reflecting the graph across the x-axis, producing the line with slope -3 and the same intercept.

ALGII141-027 5.8 Practice Problems

What happens to the parent graph 𝑦 = 3 𝑥 + 2 when the equation changes to: 𝑦 = − 3 𝑥 + 2 ?

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