Question

graphing linear equations
rodney created a graph of each given linear equation below, but rodney made mistakes on each of his graphs. use the typing tool to explain rodneys errors on the graphs he created.
for ( y = 30x + 10 ):
explain rodneys error:
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for ( y = -\frac{1}{2}x + 5 ):
explain rodneys error:
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for ( y = x - 2 ):
explain rodneys error:
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Explanation:

First Graph (\( y = 30x + 10 \))

Step1: Recall slope - intercept form

The slope - intercept form of a linear equation is \( y=mx + b \), where \( m \) is the slope and \( b \) is the y - intercept. For the equation \( y = 30x+10 \), the slope \( m = 30 \) and the y - intercept \( b = 10 \).

Step2: Analyze the graph's slope

The slope \( m=\frac{\text{rise}}{\text{run}} \). A slope of 30 means for every 1 unit increase in \( x \), \( y \) should increase by 30 units. But in the given graph, the line is decreasing (negative slope), while the slope of \( y = 30x + 10 \) is positive. So Rodney used a negative slope instead of a positive slope of 30. Also, the y - intercept seems to be at 30 (from the graph) instead of 10.

Step1: Recall slope - intercept form

For \( y =-\frac{1}{2}x + 5 \), the slope \( m=-\frac{1}{2} \) (for every 2 units increase in \( x \), \( y \) decreases by 1 unit) and y - intercept \( b = 5 \).

Step2: Analyze the graph's slope

The slope of the line in the graph is positive (the line is increasing from left to right), but the slope of \( y=-\frac{1}{2}x + 5 \) is negative. So Rodney used a positive slope instead of a negative slope of \( -\frac{1}{2} \).

Step1: Recall slope - intercept form

For a line of the form \( y=mx + b \), if the equation is \( y=x - 2 \), then \( m = 1 \) (slope) and \( b=- 2 \) (y - intercept).

Step2: Analyze the graph's y - intercept

The y - intercept of the line in the graph is at \( y=- 3 \) (or some value other than - 2) and the slope calculation: Let's check two points. If we assume the line passes through (0, - 3) and (1, - 2), the slope is 1 (correct slope), but the y - intercept is incorrect. The y - intercept of \( y=x - 2 \) should be - 2, but the graph's y - intercept is not - 2. Also, if we re - examine, the equation might be \( y=x + 2 \) by mistake, but the given equation is likely \( y=x - 2 \). So Rodney has an incorrect y - intercept.

Answer:

Rodney's error for \( y = 30x+10 \): The line has a negative slope (should be positive, \( m = 30 \)) and incorrect y - intercept (should be 10, not 30).

Second Graph (\( y=-\frac{1}{2}x + 5 \))