Comparing Linear Functions worksheet with graphs, equations, and tables for analysis.

I will solve this worksheet on "Comparing Linear Functions" by analyzing each problem step-by-step. The worksheet presents three different linear functions (A, B, and C) in various forms (graph, table, equation) and asks for their slopes, intercepts, and comparisons.

Let's break down the solution for each problem.

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Problem 1: Function A (Graph) and Function B (Table)

Function A (Graph):
- The graph is a straight line with a negative slope.
- It passes through the points (0, 3) and (3, 0).
- Slope (m): Using the formula $ m = \frac{y_2 - y_1}{x_2 - x_1} $, we get $ m = \frac{0 - 3}{3 - 0} = \frac{-3}{3} = -1 $.
- Y-intercept: The point where the line crosses the y-axis is (0, 3), so the y-intercept is 3.
- X-intercept: The point where the line crosses the x-axis is (3, 0), so the x-intercept is 3.

Function B (Table):
- The table shows:
- When $ x = 0 $, $ y = 5 $
- When $ x = 1 $, $ y = 3 $
- When $ x = 2 $, $ y = 1 $
- Slope (m): Using any two points, e.g., (0, 5) and (1, 3), we get $ m = \frac{3 - 5}{1 - 0} = \frac{-2}{1} = -2 $.
- Y-intercept: When $ x = 0 $, $ y = 5 $, so the y-intercept is 5.
- X-intercept: We can find this by setting $ y = 0 $ in the equation. First, find the equation: $ y = mx + b = -2x + 5 $. Set $ y = 0 $: $ 0 = -2x + 5 $ → $ 2x = 5 $ → $ x = 2.5 $. So the x-intercept is 2.5.

Answers for Problem 1:
- a) Which function has a greater slope? Function A has a slope of -1, while Function B has a slope of -2. Since -1 > -2, Function A has the greater slope.
- b) Which function has a greater rate of change? Rate of change is the same as slope. So, Function A has the greater rate of change.
- c) Which function has a greater y-intercept? Function A has a y-intercept of 3, Function B has a y-intercept of 5. So, Function B has the greater y-intercept.

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Problem 2: Function A (Table) and Function B (Equation)

Function A (Table):
- The table shows:
- When $ x = 0 $, $ y = -7 $
- When $ x = 1 $, $ y = -6 $
- When $ x = 2 $, $ y = -5 $
- Slope (m): Using (0, -7) and (1, -6), $ m = \frac{-6 - (-7)}{1 - 0} = \frac{1}{1} = 1 $.
- Y-intercept: When $ x = 0 $, $ y = -7 $, so the y-intercept is -7.
- X-intercept: Set $ y = 0 $ in the equation $ y = x - 7 $. $ 0 = x - 7 $ → $ x = 7 $. So the x-intercept is 7.

Function B (Equation): $ f(x) = \frac{1}{4}x - 7 $
- This is already in slope-intercept form ($ y = mx + b $).
- Slope (m): $ \frac{1}{4} $
- Y-intercept: -7
- X-intercept: Set $ y = 0 $: $ 0 = \frac{1}{4}x - 7 $ → $ \frac{1}{4}x = 7 $ → $ x = 28 $. So the x-intercept is 28.

Answers for Problem 2:
- a) Which function has a greater y-intercept? Both have a y-intercept of -7. So, neither, they are equal.
- b) Which function has a greater rate of change? Function A has a slope of 1, Function B has a slope of 1/4. Since 1 > 1/4, Function A has the greater rate of change.
- c) Which function has a greater x-intercept? Function A has an x-intercept of 7, Function B has an x-intercept of 28. So, Function B has the greater x-intercept.

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Problem 3: Function A (Graph) and Function B (Equation)

Function A (Graph):
- The graph is a straight line with a positive slope.
- It passes through the points (0, 2) and (2, 8).
- Slope (m): $ m = \frac{8 - 2}{2 - 0} = \frac{6}{2} = 3 $.
- Y-intercept: The point where the line crosses the y-axis is (0, 2), so the y-intercept is 2.
- X-intercept: Set $ y = 0 $ in the equation. First, find the equation: $ y = 3x + 2 $. Set $ y = 0 $: $ 0 = 3x + 2 $ → $ 3x = -2 $ → $ x = -\frac{2}{3} $. So the x-intercept is $ -\frac{2}{3} $.

Function B (Equation): $ f(x) = 3x + 12 $
- This is in slope-intercept form.
- Slope (m): 3
- Y-intercept: 12
- X-intercept: Set $ y = 0 $: $ 0 = 3x + 12 $ → $ 3x = -12 $ → $ x = -4 $. So the x-intercept is -4.

Answers for Problem 3:
- a) Which function has a lesser y-intercept? Function A has a y-intercept of 2, Function B has a y-intercept of 12. So, Function A has the lesser y-intercept.
- b) Which function has a greater rate of change? Both have a slope of 3. So, neither, they are equal.
- c) Which function has a greater x-intercept? Function A has an x-intercept of $ -\frac{2}{3} $, Function B has an x-intercept of -4. Since $ -\frac{2}{3} > -4 $, Function A has the greater x-intercept.

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Final Answers Summary

Here are the completed tables and answers:

Problem 1 Table:
| Function | Slope | Y-intercept | X-intercept |
|----------|-------|-------------|-------------|
| A | -1 | 3 | 3 |
| B | -2 | 5 | 2.5 |

a) Function A
b) Function A
c) Function B

Problem 2 Table:
| Function | Slope | Y-intercept | X-intercept |
|----------|-------|-------------|-------------|
| A | 1 | -7 | 7 |
| B | 1/4 | -7 | 28 |

a) Neither (equal)
b) Function A
c) Function B

Problem 3 Table:
| Function | Slope | Y-intercept | X-intercept |
|----------|-------|-------------|-------------|
| A | 3 | 2 | -2/3 |
| B | 3 | 12 | -4 |

a) Function A
b) Neither (equal)
c) Function A