Practice applying the slope-intercept form of an equation. Which graph represents the function
?
The Correct Answer and Explanation is:
y=32x−2
This equation is in slope-intercept form:y=mx+by = mx + by=mx+b
Where:
- m=23m = \frac{2}{3}m=32 is the slope
- b=−2b = -2b=−2 is the y-intercept
Step 1: Identify the y-intercept
The y-intercept tells us where the line crosses the y-axis. For this equation, the line should cross the y-axis at (0,−2)(0, -2)(0,−2). This is our starting point on the graph.
Step 2: Use the slope
The slope 23\frac{2}{3}32 means “rise over run” — for every 3 units you move to the right (positive x-direction), you go up 2 units (positive y-direction).
So, starting from (0,−2)(0, -2)(0,−2), you should move:
- Right 3 units
- Up 2 units
This should give you the next point at (3,0)(3, 0)(3,0)
Step 3: Analyze the graphs
Let’s match this to the correct graph:
- The third graph from the left:
- It has a y-intercept at -2
- From (0,−2)(0, -2)(0,−2), moving right 3 and up 2 reaches (3,0)(3, 0)(3,0)
- This matches the slope of 23\frac{2}{3}32
✅ Therefore, the third graph from the left is the correct answer.
Explanation Summary
To determine which graph represents the equation y=23x−2y = \frac{2}{3}x – 2y=32x−2, we begin by recognizing that this is written in slope-intercept form, y=mx+by = mx + by=mx+b, where mmm is the slope and bbb is the y-intercept. In this equation, the slope mmm is 23\frac{2}{3}32, and the y-intercept bbb is -2.
This means the graph should cross the y-axis at (0,−2)(0, -2)(0,−2), which is the starting point of the line. From there, we use the slope 23\frac{2}{3}32, which tells us how to move from one point to the next. A slope of 23\frac{2}{3}32 means that for every 3 units we move to the right on the x-axis, we move up 2 units on the y-axis. From (0,−2)(0, -2)(0,−2), moving right 3 units and up 2 units leads us to the point (3,0)(3, 0)(3,0), confirming the direction and steepness of the line.
Looking closely at the graphs, the third graph from the left matches both the y-intercept at -2 and the positive slope of 23\frac{2}{3}32. None of the other graphs align with both the correct slope and y-intercept. Therefore, this is the only correct representation of the equation.
Understanding how to interpret slope and y-intercept on a graph is crucial in algebra. It allows us to visually analyze relationships between variables and determine accurate graphical representations of linear equations.
