GED Math Graphs, Slope, and Functions Summary

1. Slope = (y₂ − y₁) / (x₂ − x₁).
2. Change in y: 12 − 3 = 9.
3. Change in x: 4 − 1 = 3.
4. Slope = 9 / 3 = 3.

Check: Starting at (1, 3), go right 1 and up 3 three times: (2, 6), (3, 9), (4, 12).

Wrong choices

• 1/3: 1/3 puts the change in x over the change in y; slope is rise over run.
• −3: −3 has the wrong sign; y goes up as x goes up, so the slope is positive.
• 5: 5 adds the y-values (12 + 3) instead of subtracting them.

1. Compare with y = mx + b.
2. The number multiplied by x is m = −2, the slope.
3. The number added on is b = 5, the y-intercept.

Check: When x = 0, y = −2(0) + 5 = 5, so the line crosses the y-axis at (0, 5).

Wrong choices

• Slope 5, y-intercept −2: Slope 5, y-intercept −2 switches the two numbers.
• Slope 2, y-intercept 5: Slope 2 drops the negative sign; the line goes down from left to right.
• Slope −2, y-intercept −5: The y-intercept is +5, not −5; at x = 0, y = 5.

1. In (−3, 2), the first number is x and the second is y.
2. x = −3: move 3 units to the left of the origin.
3. y = 2: move 2 units up.
4. That spot is point B.

Check: Point B is in the upper-left section, where x is negative and y is positive, which matches (−3, 2).

Wrong choices

• Point A: Point A is at (2, −3); it switches the x and y values.
• Point C: Point C is at (3, 2); its x-value is positive, not −3.
• Point D: Point D is at (−3, −2); its y-value is negative, not 2.

1. Replace x with 5: f(5) = 3(5) − 7.
2. 3 × 5 = 15.
3. 15 − 7 = 8.

Check: 8 + 7 = 15, and 15 ÷ 3 = 5, so working backward gets back to x = 5.

Wrong choices

• 22: 22 adds 7 instead of subtracting it.
• 1: 1 adds 3 and 5 instead of multiplying them (3 + 5 − 7).
• 15: 15 forgets to subtract 7.

1. Rise (change in y): 5 − (−1) = 6.
2. Run (change in x): 3 − 0 = 3.
3. Slope = 6 ÷ 3 = 2.

Check: From (0, −1), go right 1 and up 2 three times: (1, 1), (2, 3), (3, 5).

Wrong choices

• 1/2: 1/2 puts the run over the rise; slope is rise over run.
• −2: −2 has the wrong sign; the line goes up from left to right.
• 6: 6 is only the rise; it still has to be divided by the run, 3.

1. The cost per hour is the slope of the line.
2. From (0, 40) to (2, 90): rise = 90 − 40 = 50, run = 2 − 0 = 2.
3. Slope = 50 ÷ 2 = $25 per hour.

Check: From (2, 90) to (4, 140): 50 ÷ 2 = 25. The rate is the same, $25 per hour.

Wrong choices

• $40: $40 is the starting fee (the y-intercept), charged before any hours of work.
• $45: $45 divides $90 by 2 hours and includes the starting fee in the hourly rate.
• $35: $35 divides $140 by 4 hours and includes the starting fee in the hourly rate.

1. Point-slope form: y − 1 = 3(x − 2).
2. Distribute: y − 1 = 3x − 6.
3. Add 1 to both sides: y = 3x − 5.

Check: At x = 2, y = 3(2) − 5 = 1, so the line passes through (2, 1).

Wrong choices

• y = 3x + 5: y = 3x + 5 uses the wrong signs in point-slope form: y + 1 = 3(x + 2).
• y = 3x − 7: y = 3x − 7 subtracts 1 instead of adding it in the last step.
• y = 3x + 1: y = 3x + 1 uses the point's y-value as the y-intercept; at x = 2 it gives y = 7.

1. x = −4 is negative, so the point is left of the y-axis.
2. y = −6 is negative, so the point is below the x-axis.
3. Left and below is the lower-left section, Quadrant III.

Check: Quadrant I has (+, +), II has (−, +), III has (−, −), IV has (+, −).

Wrong choices

• Quadrant IV: Quadrant IV has positive x and negative y, the lower right.
• Quadrant II: Quadrant II has negative x and positive y, the upper left.
• Quadrant I: Quadrant I has positive x and positive y, the upper right.

1. Look for any x-value that appears twice with different y-values.
2. In {(1, 4), (2, 4), (3, 5)}, the x-values 1, 2, and 3 each appear once.
3. Two inputs sharing the same output (4) is allowed.
4. So this set is a function.

Check: Each x-value in this set leads to only one y-value.

Wrong choices

• {(1, 4), (1, 5), (2, 6)}: {(1, 4), (1, 5), (2, 6)} pairs x = 1 with both 4 and 5.
• {(2, 3), (3, 4), (2, 5)}: {(2, 3), (3, 4), (2, 5)} pairs x = 2 with both 3 and 5.
• {(0, 1), (4, 2), (0, 3)}: {(0, 1), (4, 2), (0, 3)} pairs x = 0 with both 1 and 3.

1. Slope: (−4 − 4) / (3 − (−1)) = −8 / 4 = −2.
2. Use (−1, 4) in y = −2x + b: 4 = −2(−1) + b = 2 + b, so b = 2.
3. Equation: y = −2x + 2.

Check: At x = 3, y = −2(3) + 2 = −4, so (3, −4) is on the line.

Wrong choices

• y = −2x + 6: y = −2x + 6 uses x = 1 instead of x = −1 when finding b; it does not pass through (−1, 4).
• y = −(1/2)x + 3.5: y = −(1/2)x + 3.5 uses run over rise for the slope.
• y = 2x + 6: y = 2x + 6 has the wrong sign on the slope; it does not pass through (3, −4).