3. Conclusion: \(g(x) = f(x + 4) = -(x + 4)\).
Teacher Note: Remind students that parallel lines NEVER represent dilations \(a \cdot f(x)\). If lines are parallel, it is always a translation! Page 1 of 3
TEACHER RESOURCE • ANSWER KEY
Graphs & Word Scenarios TEKS A.3E / AR.3A
Quick Key (7–12):
7. C 8. A 9. B 10. B 11. A 12. C
Question 7 CORRECT: C
\(g(x) = \frac{1}{4}f(x)\)
1. Intercept: Both lines pass through \((0,0)\), confirming a dilation without translation.
2. Slope Comparison: \(m_f = -2\) and \(m_g = -\frac{1}{2}\).
3. Ratio: \(a = \frac{-1/2}{-2} = \frac{1}{4}\). The line is compressed vertically (flatter).
Question 8 CORRECT: A
\(g(x) = f(x) + 4\)
1. Parallelism: Both lines have slope \(m = -2\).
2. Intercept Check: Line \(f\) passes through \((0,0)\); line \(g\) crosses at \((0,4)\).
3. Transformation: Shifted vertically upward by \(+4\) units: \(g(x) = f(x) + 4\).
Question 9 CORRECT: B
\(g(x) = f(x - 2)\)
1. Horizontal Shift: The x-intercept shifted right from \((0,0)\) to \((2,0)\).
2. Function Rule: Right shift of 2 units means substituting \((x - 2)\) into \(f(x)\).
3. Check: \(g(x) = -(x - 2) = -x + 2\), matching the graph exactly.
Question 10 CORRECT: B
\(g(x) = 4f(x)\)
1. Intercept: Intersects at \((0,0)\); represents a pure dilation.
2. Slopes: Line \(f\) has slope \(\frac{1}{2}\); line \(g\) has slope \(2\).
3. Scale Factor: \(a = \frac{2}{1/2} = 4\). Line \(g\) is vertically stretched (4 times steeper).
Question 11 (Hourly Wage) CORRECT: A
Vertical stretch (steeper) by factor 2
Context: Hourly rate increases from $\$30/\text{hr}$ to $\$60/\text{hr}$.
Dilation Factor: \(g(x) = 2f(x)\), where scale factor \(a = 2 > 1\).
Geometric Effect: The slope doubles from 30 to 60, making the graph steeper.
Question 12 (Water Drainage) CORRECT: C
Vertical compression (flatter) by factor \(\frac{1}{4}\)
Context: Discharge rate drops from \(-80\,\text{gal/hr}\) to \(-20\,\text{gal/hr}\).
Dilation Factor: \(g(x) = \frac{1}{4}f(x)\), where scale factor \(a = \frac{1}{4} < 1\).
Geometric Effect: Absolute slope \(|m|\) decreases from 80 to 20 (line becomes flatter).
Key Takeaway: When comparing negative slopes, remember that steepness corresponds to \(|m|\). A slope of \(-80\) is much steeper than a slope of \(-20\)! Page 2 of 3
TEACHER RESOURCE • ANSWER KEY
Contexts & Analysis TEKS A.3E / AR.3A
Quick Key (13–18):
13. A 14. A 15. A 16. A 17. A 18. A
Q13 (Temperature Drop) A
Shift UP 12: \(g(x) = f(x) + 12\). Adding a positive constant outside the function translates every y-value vertically upward by 12 units.
Q14 (Rideshare Net) A
Shift DOWN 20: \(g(x) = f(x) - 20\). Subtracting 20 reduces total earnings uniformly across all miles, shifting the graph down 20 units.
Q15 (Bakery Production) A
Shift RIGHT 2: Starting 2 hours later means the zero point shifts from \(x = 0\) to \(x = 2\). In function notation, right by 2 is \(f(x - 2)\).
Q16 (Aircraft Descent) A
Shift LEFT 5: Initiating descent 5 minutes earlier shifts the timeline backward to \(x = -5\). In function notation, left by 5 is \(f(x + 5)\).
Q17 (Snowpack Melt) A
Steepness = \(|m|\): While \(-4.5 < -1.5\) on a number line, geometric steepness is measured by magnitude \(|-4.5| = 4.5 > 1.5\).
Q18 (Battery Saver) A
Flatter Slope & Same Intercept: Multiplying by \(\frac{1}{3}\) reduces the absolute rate of discharge from 18 to 6, flattening the line while keeping \((0,0)\).
COMMON MISCONCEPTIONS & TARGETED INTERVENTIONS
Misconception 1: Confusing Dilations with Shifts
Students see a flatter line below the original and assume it was translated downward (\(f(x) - d\)).
Intervention: Point to \((0,0)\). A vertical shift changes the y-intercept. If both lines cross \((0,0)\), it CANNOT be a vertical shift!
Misconception 2: Horizontal Translation Signs
Students often believe \(f(x - 3)\) moves left because minus indicates negative direction.
Intervention: Solve for the new zero: \(x - 3 = 0 \implies x = +3\). To reach zero, \(x\) must increase by 3 (shift RIGHT).
Question 7 • \(g(x) = \frac{1}{4}f(x)\) Correct: C
Both cross \((0,0)\). Slope \(m_f = -2\); slope \(m_g = -\frac{1}{2}\). Scale factor \(a = \frac{-1/2}{-2} = \frac{1}{4}\). The line is vertically compressed.
Distractor: Option A (\(4f(x)\)) would produce a much steeper line of slope \(-8\).
Question 8 • \(g(x) = f(x) + 4\) Correct: A
Lines are parallel (\(m = -2\)). The y-intercept translates from \((0,0)\) up to \((0,4)\). Therefore, \(g(x) = f(x) + 4\).
Distractor: Even with negative slope, adding \(+4\) outside moves the line vertically up.
Question 9 • \(g(x) = f(x - 2)\) Correct: B
Lines are parallel (\(m = -1\)). The x-intercept shifts from \((0,0)\) to \((2,0)\) (2 units right). Horizontal translation right: \(f(x - 2)\).
Distractor: Note that for \(m = -1\), \(f(x - 2) = -(x - 2) = -x + 2\), shifting the y-intercept to \(+2\).
Question 10 • \(g(x) = 4f(x)\) Correct: B
Both lines pass through \((0,0)\). Slope \(m_f = \frac{1}{2}\); slope \(m_g = 2\). Scale factor \(a = \frac{2}{1/2} = 4\). Vertically stretched by 4.
Distractor: Students subtracting intercepts guess \(f(x) + 2\); check that both intersect at origin.
3 High-Impact Diagnostic Rules for Teachers
1. Origin Anchor Test If both lines pass through \((0,0)\), it is ALWAYS a dilation: \(a = \frac{m_g}{m_f}\). It cannot be a vertical or horizontal translation.
2. Negative Slope Steepness Remind students that steepness is \(|m|\). A line with slope \(-3\) is STEEPER than \(-1\) because \(|-3| > |-1|\). The dilation factor is still positive!
3. Sign Reversal for Horizontal Outside the function (\(f(x) \pm d\)) moves vertically (\(+d\) up, \(-d\) down). Inside the function (\(f(x \pm c)\)) moves horizontally (\(-c\) right, \(+c\) left).
