Transformations of Functions: Algebra 2 Math Worksheet – PDF Download

Problem 1: Describe the shift

Describe how each graph is related to the graph of \(y=x^{2}\).

1. \(g(x)=x^{2}+5\)
2. \(h(x)=x^{2}-3\)
3. \(j(x)=(x-4)^{2}\)
4. \(k(x)=(x+1)^{2}\)

Problem 2: Absolute value from a graph

The graph of an absolute value function is shown.

1. Give the vertex.
2. Write an equation of the form \(y=|x-h|+k\).

Problem 3: Transform a table

The table shows values of \(f(x)\). Make a table for each new function.

\(x\)	\(-2\)	\(-1\)	\(0\)	\(1\)	\(2\)
\(f(x)\)	\(3\)	\(-1\)	\(2\)	\(5\)	\(0\)

1. \(f(x)+4\)
2. \(f(x-1)\)
3. \(2f(x)\)
4. \(-f(x)\)

Problem 4: Reflections

Describe how each graph is related to the graph of \(y=\sqrt{x}\), and state the domain of each.

1. \(g(x)=-\sqrt{x}\)
2. \(h(x)=\sqrt{-x}\)

Problem 5: Stretch or compress

Describe how each graph is related to \(y=|x|\).

1. \(g(x)=3|x|\)
2. \(h(x)=\frac{1}{3}|x|\)
3. \(k(x)=|3x|\)

Problem 6: Write the equation

Write an equation for the graph after the transformation.

1. \(y=x^{2}\) shifted left 3 and down 6
2. \(y=|x|\) shifted right 5 and up 1
3. \(y=\sqrt{x}\) shifted down 4
4. \(y=2^{x}\) shifted up 3

Problem 7: Several transformations

Describe every transformation of \(y=x^{2}\) that produces \(g(x)=-2(x-1)^{2}+5\). State the vertex and whether the parabola opens up or down.

Problem 8: Exponential transformations

The graph of the parent function \(f(x)=2^{x}\) is shown. Let \(g(x)=2^{x}-3\) and \(h(x)=2^{x+1}\).

1. Give the horizontal asymptote and the y-intercept of \(g\).
2. Give the horizontal asymptote and the y-intercept of \(h\).
3. Show that \(h(x)=2\cdot 2^{x}\).

Problem 9: Even, odd or neither

Use \(f(-x)\) to decide whether each function is even, odd or neither.

1. \(f(x)=x^{4}-3x^{2}\)
2. \(g(x)=x^{3}+x\)
3. \(h(x)=x^{2}+x\)

Problem 10: Use symmetry

The function \(f\) is odd with \(f(1)=3\), \(f(2)=-5\) and \(f(3)=8\). The function \(g\) is even with \(g(2)=7\).

1. Find \(f(-1)\), \(f(-2)\), \(f(-3)\) and \(f(0)\).
2. Find \(g(-2)\).

Problem 11: Parent functions and symmetry

Classify each parent function as even, odd or neither, and explain.

1. \(y=|x|\)
2. \(y=\sqrt[3]{x}\)
3. \(y=\sqrt{x}\)
4. \(y=2^{x}\)

Problem 12: Graph a square root function

Graph \(y=\sqrt{x-2}+1\). Then state the domain and range.

Problem 13: Graph a cube root function

Graph \(y=\sqrt[3]{x+1}-2\) by finding five points, one of them the center. State the domain and range.

Problem 14: Read a square root graph

The graph of a square root function is shown.

1. State the domain and the range.
2. Write the equation in the form \(y=a\sqrt{x-h}+k\).

Problem 15: Read a cube root graph

Write an equation of the form \(y=\sqrt[3]{x-h}+k\) for the graph.

Problem 16: Move the points

The point \((6,-4)\) is on the graph of \(y=f(x)\). Find the corresponding point on each graph.

1. \(y=f(x)+3\)
2. \(y=f(x-2)\)
3. \(y=2f(x)\)
4. \(y=f(3x)\)
5. \(y=f(-x)\)
6. \(y=-f(x)\)

Problem 17: Ball toss

Noah tosses a ball straight up from the ground. Its height in feet after \(t\) seconds is \(h(t)=-16(t-2)^{2}+64\).

1. Describe the transformations of \(y=t^{2}\).
2. Find the maximum height and when it happens.
3. Find \(h(0)\) and \(h(4)\), and tell how long the ball is in the air.

Problem 18: Weekly profit

The school store’s weekly profit in dollars from selling \(x\) hoodies is \(P(x)=-2(x-15)^{2}+450\).

1. Find the maximum profit and the number of hoodies that produces it.
2. Find \(P(10)\) and \(P(20)\). Why are they equal?

Problem 19: Gift box

Aisha packs a gift in a cube-shaped box with volume \(V\) cubic inches. The edge length of the outer shipping box is \(E(V)=\sqrt[3]{V}+2\) inches, which allows 1 inch of padding on each side.

1. Describe how the graph of \(E\) is related to \(y=\sqrt[3]{V}\).
2. Find \(E(27)\) and \(E(125)\).
3. Find the gift volume when the shipping box edge is 9 inches.

Problem 20: Skid marks

Police estimate a car’s speed in miles per hour from the length \(d\) in feet of its skid marks with \(s(d)=\sqrt{24d}\).

1. Find the speed for skid marks of 6 feet, 54 feet and 150 feet.
2. Describe \(s\) as a transformation of \(y=\sqrt{d}\).
3. How long are the skid marks of a car going 48 mph?

Problem 21: Bacteria colony

Mei’s science project grows a colony of \(N(t)=500\cdot 2^{t}\) bacteria after \(t\) hours.

1. Find \(N(0)\) and \(N(3)\).
2. Write \(N(t+2)\) and show that it equals \(4N(t)\).
3. Explain in words why shifting the graph left 2 hours gives the same graph as stretching it vertically by a factor of 4.

Problem 22: Thermostat error

Liam’s thermostat is set to 68°F. The error when the room temperature is \(T\) degrees is \(E(T)=|T-68|\).

1. Describe how the graph of \(E\) is related to \(y=|T|\).
2. Find \(E(65)\) and \(E(71)\).
3. At which temperatures is the error exactly 5 degrees?

See every chapter on the Algebra 2 math page.