This algebra 2 transformations worksheet answer key solves all 22 problems of the transformations of functions worksheet step by step.
Each solution shows which part of the equation causes which change, and the graphs mark the key points. If an answer differs, check whether a shift inside the function was applied in the wrong direction.
Algebra 2 Transformations Worksheet Answer Key: problems 1–5
Problem 1: Describe the shift
- Up 5.
- Down 3.
- Right 4 (the \(-4\) inside moves the graph right).
- Left 1.
Answers: \(\text{up }5,\ \text{down }3,\ \text{right }4,\ \text{left }1\).
Problem 2: Absolute value from a graph
- The vertex is \((-2,-3)\).
- \(h=-2\) and \(k=-3\): \(y=|x+2|-3\). Check \((1,0)\): \(|3|-3=0\).
Problem 3: Transform a table
- Add 4 to each output: \((-2,7),(-1,3),(0,6),(1,9),(2,4)\).
- Move each point right 1: \((-1,3),(0,-1),(1,2),(2,5),(3,0)\).
- Double each output: \((-2,6),(-1,-2),(0,4),(1,10),(2,0)\).
- Change each output’s sign: \((-2,-3),(-1,1),(0,-2),(1,-5),(2,0)\).
Problem 4: Reflections
- Multiplying the output by \(-1\) reflects the graph over the x-axis. Domain: \(x\ge 0\).
- Replacing \(x\) with \(-x\) reflects the graph over the y-axis. We need \(-x\ge 0\), so the domain is \(x\le 0\).
Problem 5: Stretch or compress
- Vertical stretch by a factor of 3.
- Vertical compression by a factor of \(\frac{1}{3}\).
- \(|3x|=3|x|\), so it is the same as part (a): a horizontal compression by \(\frac{1}{3}\), which looks like a vertical stretch by 3.
Algebra 2 Transformations Worksheet Answer Key: problems 6–10
Problem 6: Write the equation
- \(y=(x+3)^{2}-6\)
- \(y=|x-5|+1\)
- \(y=\sqrt{x}-4\)
- \(y=2^{x}+3\)
Problem 7: Several transformations
Factor \(2\): vertical stretch by 2. The negative sign: reflection over the x-axis. \(x-1\): right 1. \(+5\): up 5. The vertex is \((1,5)\) and the parabola opens down.
Problem 8: Exponential transformations
- The graph moves down 3, so the asymptote is \(y=-3\). \(g(0)=1-3=-2\): y-intercept \((0,-2)\).
- A horizontal shift does not move the asymptote: \(y=0\). \(h(0)=2^{1}=2\): \((0,2)\).
- \(2^{x+1}=2^{x}\cdot 2^{1}=2\cdot 2^{x}\), so shifting left 1 is a vertical stretch by 2.
Problem 9: Even, odd or neither
- \(f(-x)=(-x)^{4}-3(-x)^{2}=x^{4}-3x^{2}=f(x)\): \(\text{even}\).
- \(g(-x)=-x^{3}-x=-(x^{3}+x)=-g(x)\): \(\text{odd}\).
- \(h(-x)=x^{2}-x\), which is neither \(h(x)\) nor \(-h(x)\): \(\text{neither}\).
Problem 10: Use symmetry
- For an odd function \(f(-x)=-f(x)\): \(f(-1)=-3\), \(f(-2)=5\), \(f(-3)=-8\). Also \(f(0)=-f(0)\), so \(f(0)=0\).
- For an even function \(g(-2)=g(2)=7\).
Algebra 2 Transformations Worksheet Answer Key: problems 11–15
Problem 11: Parent functions and symmetry
- \(|-x|=|x|\): \(\text{even}\).
- \(\sqrt[3]{-x}=-\sqrt[3]{x}\): \(\text{odd}\).
- \(\sqrt{-1}\) is not real, so \(f(-x)\) is not defined for positive \(x\): neither.
- \(2^{-1}=\frac{1}{2}\), which is neither \(2\) nor \(-2\): neither.
Problem 12: Graph a square root function
The graph starts at \((2,1)\). Choose \(x-2\) to be 0, 1, 4 and 9: \((2,1),(3,2),(6,3),(11,4)\).
Domain: \(x\ge 2\). Range: \(y\ge 1\).
Problem 13: Graph a cube root function
The center is \((-1,-2)\). Choose \(x+1\) to be \(-8,-1,0,1,8\): \(x=-9,-2,-1,0,7\) give \(y=-4,-3,-2,-1,0\).
Domain: all real numbers. Range: all real numbers.
Problem 14: Read a square root graph
- It starts at \((-3,2)\) and goes right and down. Domain: \(x\ge -3\). Range: \(y\le 2\).
- \(h=-3\), \(k=2\). The point \((1,0)\) gives \(0=a\sqrt{4}+2\), so \(a=-1\). Equation: \(y=-\sqrt{x+3}+2\). Check \((6,-1)\): \(-3+2=-1\).
Problem 15: Read a cube root graph
The graph flattens at its center \((2,-1)\), so \(h=2\) and \(k=-1\). Check \((10,1)\): \(\sqrt[3]{8}-1=1\), and \((-6,-3)\): \(\sqrt[3]{-8}-1=-3\). Equation: \(y=\sqrt[3]{x-2}-1\).
Algebra 2 Transformations Worksheet Answer Key: problems 16–20
Problem 16: Move the points
- Add 3 to the y-coordinate: \((6,-1)\).
- Add 2 to the x-coordinate: \((8,-4)\).
- Double the y-coordinate: \((6,-8)\).
- Divide the x-coordinate by 3: \((2,-4)\).
- Negate the x-coordinate: \((-6,-4)\).
- Negate the y-coordinate: \((6,4)\).
Problem 17: Ball toss
- Vertical stretch by 16, reflection over the t-axis, right 2, up 64.
- The vertex is \((2,64)\): the maximum height is 64 feet at 2 seconds.
- \(h(0)=-16(4)+64=0\) and \(h(4)=-16(4)+64=0\). The ball is in the air 4 seconds.
Problem 18: Weekly profit
- The vertex is \((15,450)\): the maximum profit is $450 with 15 hoodies.
- \(P(10)=-2(25)+450=400\) and \(P(20)=-2(25)+450=400\). Both are 5 units from 15, and the parabola is symmetric about \(x=15\).
Problem 19: Gift box
- It is shifted up 2.
- \(E(27)=3+2=5\) and \(E(125)=5+2=7\). Edges: 5 in. and 7 in.
- \(\sqrt[3]{V}+2=9\) gives \(\sqrt[3]{V}=7\) and \(V=343\). The gift volume is 343 cubic inches.
Problem 20: Skid marks
- \(\sqrt{144}=12\), \(\sqrt{1296}=36\), \(\sqrt{3600}=60\). Speeds: 12, 36 and 60 mph.
- \(s(d)=f(24d)\) for \(f(d)=\sqrt{d}\): a horizontal compression by \(\frac{1}{24}\).
- \(48=\sqrt{24d}\) gives \(2304=24d\) and \(d=96\). The marks are 96 feet long.
Algebra 2 Transformations Worksheet Answer Key: problems 21–22
Problem 21: Bacteria colony
- \(N(0)=500\) and \(N(3)=500\cdot 8=4000\).
- \(N(t+2)=500\cdot 2^{t+2}=500\cdot 4\cdot 2^{t}=2000\cdot 2^{t}=4N(t)\).
- Waiting 2 more hours quadruples the colony at every time, so every height on the left-shifted graph is 4 times the original height.
Problem 22: Thermostat error
- It is shifted right 68, so the vertex is at \((68,0)\).
- \(E(65)=|-3|=3\) and \(E(71)=|3|=3\).
- \(|T-68|=5\) gives \(T-68=5\) or \(T-68=-5\), so \(T=73\) or \(T=63\). Temperatures: 63°F and 73°F.
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Algebra 2 Transformations of Functions worksheet answer key
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