Which equation represents the graphed function? y = -3x + 3 y = 3x – 3 y = 3x – \frac{1}{3} y = -\frac{1}{3}x + 3
The Correct Answer and Explanation is:
To find which equation represents the graphed function, we need to identify two key features from the graph:
1. Y-Intercept
From the graph, the line passes through the point (0,3)(0, 3). This is the y-intercept, which tells us the value of yy when x=0x = 0.
So, the equation must be of the form: y=mx+3y = mx + 3
This eliminates any options without “+ 3” at the end. We are now left with:
- y=−3x+3y = -3x + 3
- y=−13x+3y = -\frac{1}{3}x + 3
2. Slope (m)
The line also passes through the point (3,2)(3, 2).
We can calculate the slope mm using the slope formula: m=y2−y1x2−x1=2−33−0=−13m = \frac{y_2 – y_1}{x_2 – x_1} = \frac{2 – 3}{3 – 0} = \frac{-1}{3}
So, the slope is −13-\frac{1}{3}.
Now plug that slope into the equation with the correct y-intercept: y=−13x+3y = -\frac{1}{3}x + 3
Thus, the correct answer is:
D) y=−13x+3y = -\frac{1}{3}x + 3
Explanation
The graph represents a linear function, and the equation of a line is generally written in slope-intercept form: y=mx+by = mx + b
Where:
- mm is the slope (the rate of change of y with respect to x),
- bb is the y-intercept (the point where the line crosses the y-axis).
From the graph, we can see that the line crosses the y-axis at y=3y = 3, so b=3b = 3. That means any correct equation must end with +3. This already eliminates two of the answer choices.
Next, to find the slope, we identify two points the line passes through: (0,3)(0, 3) and (3,2)(3, 2).
Using the slope formula: m=change in ychange in x=2−33−0=−13m = \frac{\text{change in } y}{\text{change in } x} = \frac{2 – 3}{3 – 0} = \frac{-1}{3}
The slope is negative, meaning the line is decreasing, which matches the graph.
So the correct equation becomes: y=−13x+3y = -\frac{1}{3}x + 3
This matches answer choice D, making it the correct representation of the graphed function.
