Problem 1: Shift down a parabola
Describe how the graph of \(g(x)=x^{2}-6\) compares with the graph of \(f(x)=x^{2}\). Give the vertex of \(g\).
Problem 2: Shift up an absolute value
Describe how the graph of \(g(x)=|x|+7\) compares with the graph of \(f(x)=|x|\). Give the vertex of \(g\).
Problem 3: Equation from a shifted parabola
The dashed graph is \(f(x)=x^{2}\). The solid graph \(g\) has the same shape. Write \(g(x)\) in terms of \(f\) and then as an equation in \(x\).
Problem 4: Shift left
Describe how the graph of \(g(x)=|x+3|\) compares with \(f(x)=|x|\). Give the vertex and the y-intercept of \(g\).
Problem 5: Add outside or inside
Let \(f(x)=x^{2}\). Evaluate \(f(3)+2\) and \(f(3+2)\). Explain why the results differ.
Problem 6: Stretch table
Copy and complete the table for \(f(x)=|x|\) and \(g(x)=3|x|\). Then describe how the graph of \(g\) compares with the graph of \(f\).
| \(x\) | \(-2\) | \(-1\) | \(0\) | \(1\) | \(2\) |
|---|---|---|---|---|---|
| \(f(x)\) | |||||
| \(g(x)\) |
Problem 7: Equation from an absolute value graph
The graph of \(g\) is a shifted copy of \(f(x)=|x|\). Write the equation of \(g\) in the form \(g(x)=|x-h|+k\).
Problem 8: Reflect and shrink
Write \(g(x)\) in terms of \(f(x)=x^{2}\), then as an equation in \(x\), and describe the transformation.
Problem 9: Transform a linear function
Let \(f(x)=2x+1\). Write each function in simplified form: (a) \(g(x)=f(x)+3\); (b) \(h(x)=f(x+3)\); (c) \(k(x)=-f(x)\).
Problem 10: Vertex and range
For \(g(x)=|x-4|+2\), find the vertex, the equation of the axis of symmetry, the minimum value, the domain and the range.
Problem 11: Graph an absolute value function
Make a table of values for \(x=-2,-1,0,1,2,3,4\) and graph \(y=|x-1|-2\). Label the vertex.
Problem 12: Intercepts of an absolute value graph
Find the y-intercept and the x-intercepts of \(g(x)=2|x+1|-4\).
Problem 13: An absolute value that opens down
For \(h(x)=-|x-3|+5\), find the vertex, the maximum value, the y-intercept and the x-intercepts.
Problem 14: Evaluate a piecewise function
Let \(f(x)=x+4\) when \(x < 0\), and \(f(x)=x^{2}\) when \(x\ge 0\). Find \(f(-3)\), \(f(-1)\), \(f(0)\) and \(f(2)\).
Problem 15: Graph a piecewise function
Graph \(f(x)=\frac{1}{2}x+1\) when \(x < 2\), and \(f(x)=-x+5\) when \(x\ge 2\), for \(-4\le x\le 6\). Use an open or closed circle at \(x=2\) for each piece.
Problem 16: Write a piecewise function
Write a piecewise function for the graph. Then find \(f(-4)\), \(f(3)\) and \(f(5)\).
Problem 17: Greatest integer step function
Let \(s(x)\) be the greatest integer less than or equal to \(x\). Find \(s(3.7)\), \(s(-1.2)\), \(s(5)\), \(s(-4)\) and \(s(0.99)\).
Problem 18: Check a point
Is the point \((4,0)\) on the graph of \(g(x)=-2|x-1|+6\)? Is the point \((-2,2)\)?
Problem 19: Ride-share booking fee
Maya’s ride costs \(f(m)=2m\) dollars for \(m\) miles. The company adds a $4 booking fee, so the new cost is \(g(m)=f(m)+4\). Describe the transformation of the graph, write \(g(m)\), and find the cost of a 7-mile ride.
Problem 20: Starting to save later
Jamal saves $15 each week, so after \(w\) weeks he has \(f(w)=15w\) dollars. This year he starts saving 3 weeks later. Write the new function \(g\) in terms of \(f\), describe the shift, and find how much he has saved after week 8.
Problem 21: Dropped ball
A ball is dropped from a platform. Its height in feet after \(t\) seconds is \(h(t)=-16t^{2}+64\). Describe how the graph of \(h\) is obtained from \(f(t)=t^{2}\), give the starting height, and find when the ball lands.
Problem 22: Data plan
A phone plan costs $20 per month for up to 5 GB of data, plus $4 for each additional GB. Let \(C(g)\) be the monthly cost for \(g\) GB. Write \(C(g)\) as a piecewise function, then find the cost for 3 GB, 8 GB and 12 GB, and graph the function.
Problem 23: Parking garage
A garage charges $5 for each hour or part of an hour. The graph shows the cost. Find the cost of parking for (a) 2.3 hours, (b) exactly 3 hours, (c) 3.01 hours.
Problem 24: Bottle tolerance
A machine fills bottles to \(x\) ounces. The error is \(E(x)=|x-16|\), and a bottle is acceptable when \(E(x)\le 0.2\). Is a bottle with 15.9 ounces acceptable? Is one with 16.3 ounces? Write the acceptable amounts as an inequality.
See every chapter on the Algebra 1 math page.