Transformations of Functions: Algebra 2 Math Worksheet Answer Key – PDF Download

Problem 1: Describe the shift

1. Up 5.
2. Down 3.
3. Right 4 (the \(-4\) inside moves the graph right).
4. Left 1.

Answers: \(\text{up }5,\ \text{down }3,\ \text{right }4,\ \text{left }1\).

Problem 2: Absolute value from a graph

1. The vertex is \((-2,-3)\).
2. \(h=-2\) and \(k=-3\): \(y=|x+2|-3\). Check \((1,0)\): \(|3|-3=0\).

Problem 3: Transform a table

1. Add 4 to each output: \((-2,7),(-1,3),(0,6),(1,9),(2,4)\).
2. Move each point right 1: \((-1,3),(0,-1),(1,2),(2,5),(3,0)\).
3. Double each output: \((-2,6),(-1,-2),(0,4),(1,10),(2,0)\).
4. Change each output’s sign: \((-2,-3),(-1,1),(0,-2),(1,-5),(2,0)\).

Problem 4: Reflections

1. Multiplying the output by \(-1\) reflects the graph over the x-axis. Domain: \(x\ge 0\).
2. 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

1. Vertical stretch by a factor of 3.
2. Vertical compression by a factor of \(\frac{1}{3}\).
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.

Problem 6: Write the equation

1. \(y=(x+3)^{2}-6\)
2. \(y=|x-5|+1\)
3. \(y=\sqrt{x}-4\)
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

1. The graph moves down 3, so the asymptote is \(y=-3\). \(g(0)=1-3=-2\): y-intercept \((0,-2)\).
2. A horizontal shift does not move the asymptote: \(y=0\). \(h(0)=2^{1}=2\): \((0,2)\).
3. \(2^{x+1}=2^{x}\cdot 2^{1}=2\cdot 2^{x}\), so shifting left 1 is a vertical stretch by 2.

Problem 9: Even, odd or neither

1. \(f(-x)=(-x)^{4}-3(-x)^{2}=x^{4}-3x^{2}=f(x)\): \(\text{even}\).
2. \(g(-x)=-x^{3}-x=-(x^{3}+x)=-g(x)\): \(\text{odd}\).
3. \(h(-x)=x^{2}-x\), which is neither \(h(x)\) nor \(-h(x)\): \(\text{neither}\).

Problem 10: Use symmetry

1. For an odd function \(f(-x)=-f(x)\): \(f(-1)=-3\), \(f(-2)=5\), \(f(-3)=-8\). Also \(f(0)=-f(0)\), so \(f(0)=0\).
2. For an even function \(g(-2)=g(2)=7\).

Problem 11: Parent functions and symmetry

1. \(|-x|=|x|\): \(\text{even}\).
2. \(\sqrt[3]{-x}=-\sqrt[3]{x}\): \(\text{odd}\).
3. \(\sqrt{-1}\) is not real, so \(f(-x)\) is not defined for positive \(x\): neither.
4. \(2^{-1}=\frac{1}{2}\), 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

1. It starts at \((-3,2)\) and goes right and down. Domain: \(x\ge -3\). Range: \(y\le 2\).
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: \(y=\sqrt[3]{x-2}-1\).

Problem 16: Move the points

1. Add 3 to the y-coordinate: \((6,-1)\).
2. Add 2 to the x-coordinate: \((8,-4)\).
3. Double the y-coordinate: \((6,-8)\).
4. Divide the x-coordinate by 3: \((2,-4)\).
5. Negate the x-coordinate: \((-6,-4)\).
6. Negate the y-coordinate: \((6,4)\).

Problem 17: Ball toss

1. Vertical stretch by 16, reflection over the t-axis, right 2, up 64.
2. The vertex is \((2,64)\): the maximum height is 64 feet at 2 seconds.
3. \(h(0)=-16(4)+64=0\) and \(h(4)=-16(4)+64=0\). The ball is in the air 4 seconds.

Problem 18: Weekly profit

1. The vertex is \((15,450)\): the maximum profit is $450 with 15 hoodies.
2. \(P(10)=-2(25)+450=400\) 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

1. It is shifted up 2.
2. \(E(27)=3+2=5\) and \(E(125)=5+2=7\). Edges: 5 in. and 7 in.
3. \(\sqrt[3]{V}+2=9\) gives \(\sqrt[3]{V}=7\) and \(V=343\). The gift volume is 343 cubic inches.

Problem 20: Skid marks

1. \(\sqrt{144}=12\), \(\sqrt{1296}=36\), \(\sqrt{3600}=60\). Speeds: 12, 36 and 60 mph.
2. \(s(d)=f(24d)\) for \(f(d)=\sqrt{d}\): a horizontal compression by \(\frac{1}{24}\).
3. \(48=\sqrt{24d}\) gives \(2304=24d\) and \(d=96\). The marks are 96 feet long.

Problem 21: Bacteria colony

1. \(N(0)=500\) and \(N(3)=500\cdot 8=4000\).
2. \(N(t+2)=500\cdot 2^{t+2}=500\cdot 4\cdot 2^{t}=2000\cdot 2^{t}=4N(t)\).
3. 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

1. It is shifted right 68, so the vertex is at \((68,0)\).
2. \(E(65)=|-3|=3\) and \(E(71)=|3|=3\).
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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