The worksheet provided is an
Answer Key, meaning the solutions are already graphed and labeled on the coordinate planes. Below is the verification of each problem to ensure the graphs match the equations.
Step-by-Step Verification
1. Equation: $y = -\frac{1}{2}x + 5$
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Identify parts: The slope ($m$) is $-\frac{1}{2}$ and the y-intercept ($b$) is $5$.
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Check Point 1 (Y-intercept): Start at $(0, 5)$. The graph shows a point at $(0, 5)$. This is correct.
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Check Point 2: Use the slope "down 1, right 2" from $(0,5)$.
* Down 1 from $y=5$ is $4$. Right 2 from $x=0$ is $2$. New point: $(2, 4)$.
* Repeat: Down 1 to $3$, Right 2 to $4$. New point: $(4, 3)$.
* Repeat: Down 1 to $2$, Right 2 to $6$. New point: $(6, 2)$.
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Result: The graph correctly plots the line passing through $(0,5)$, $(2,4)$, $(4,3)$, and $(6,2)$.
2. Equation: $y = \frac{3}{2}x + 1$
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Identify parts: The slope ($m$) is $\frac{3}{2}$ and the y-intercept ($b$) is $1$.
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Check Point 1 (Y-intercept): Start at $(0, 1)$. The graph shows a point at $(0, 1)$. This is correct.
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Check Point 2: Use the slope "up 3, right 2".
* Up 3 from $y=1$ is $4$. Right 2 from $x=0$ is $2$. New point: $(2, 4)$.
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Check Point 3: Go backwards using "down 3, left 2" from $(0,1)$.
* Down 3 from $y=1$ is $-2$. Left 2 from $x=0$ is $-2$. New point: $(-2, -2)$.
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Result: The graph correctly plots the line passing through $(-2,-2)$, $(0,1)$, and $(2,4)$.
3. Equation: $y = x$
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Identify parts: This can be written as $y = 1x + 0$. Slope is $1$, y-intercept is $0$.
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Check Point 1 (Origin): The line passes through $(0,0)$. This is correct.
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Check Point 2: Slope is $1$ ("up 1, right 1").
* From $(0,0)$, go up 1 to $1$, right 1 to $1$. Point: $(1, 1)$.
* From $(1,1)$, go up 1 to $2$, right 1 to $2$. Point: $(2, 2)$.
* From $(2,2)$, go up 1 to $3$, right 1 to $3$. Point: $(3, 3)$.
* From $(3,3)$, go up 1 to $4$, right 1 to $4$. Point: $(4, 4)$.
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Result: The graph correctly plots the diagonal line passing through $(0,0), (1,1), (2,2), (3,3),$ and $(4,4)$.
4. Equation: $x = -5$
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Identify parts: This is a vertical line where the x-coordinate is always $-5$, regardless of the y-value.
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Check Points:
* Point at $y=0$: $(-5, 0)$.
* Point at $y=5$: $(-5, 5)$.
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Result: The graph correctly shows a vertical line crossing the x-axis at $-5$ and passing through $(-5,0)$ and $(-5,5)$.
Final Answer:
The graphs on the worksheet are correct. They accurately represent the following linear equations:
1.
$y = -\frac{1}{2}x + 5$: A line with a y-intercept of 5 and a negative slope.
2.
$y = \frac{3}{2}x + 1$: A line with a y-intercept of 1 and a positive steep slope.
3.
$y = x$: A diagonal line passing through the origin $(0,0)$ with a slope of 1.
4.
$x = -5$: A vertical line crossing the x-axis at -5.
Parent Tip: Review the logic above to help your child master the concept of worksheet graphing linear equations.
