These 22 problems practice F-IF.B.4, F-IF.B.6 and F-IF.C.9. You will find intercepts, intervals of increase and decrease, maximum and minimum values, end behavior and symmetry, then compute average rates of change and compare functions given by equations, tables and graphs.
Problems move from reading simple graphs to word problems about speed, tides, tanks and memberships. Always include units when the context has them.
Problem 1: Intercepts of a line
Find the \(x\)-intercept and the \(y\)-intercept of \(f(x)=2x-6\).
Problem 2: Intercepts of a parabola
Find the intercepts of \(g(x)=x^{2}-9\).
Problem 3: Features from a graph
Use the graph to find the \(x\)-intercepts, the \(y\)-intercept, the vertex and the axis of symmetry. Is the vertex a maximum or a minimum?
Problem 4: Intercepts from a table
The table shows a linear function \(f\). Find the \(y\)-intercept and the \(x\)-intercept. Is \(f\) increasing or decreasing?
Problem 5: Increasing and decreasing
Give the intervals where the function is increasing and decreasing, and give the relative maximum and relative minimum.
Problem 6: Positive and negative intervals
The graph shows a function \(f\). Find the \(x\)-intercepts, the interval where \(f\) is positive, and the interval where \(f\) is negative.
Problem 7: Axis of symmetry and vertex
For \(f(x)=x^{2}-6x+5\), find the axis of symmetry and the vertex. Then find the point symmetric to the \(y\)-intercept.
Problem 8: Features of an absolute value function
List the vertex, the intercepts, the intervals of increase and decrease, and the line of symmetry for \(f(x)=|x-2|-3\).
Problem 9: Maximum of a quadratic
For \(f(x)=-x^{2}+4x+1\), find the vertex, the maximum value, the \(y\)-intercept and the intervals of increase and decrease.
Problem 10: End behavior
Describe the end behavior of each function: what happens to \(f(x)\) as \(x\) gets very large, and as \(x\) gets very negative?
- \(f(x)=-2x+5\)
- \(g(x)=x^{2}+1\)
- \(h(x)=3^{x}\)
Problem 11: Average rate of change: squares
Find the average rate of change of \(f(x)=x^{2}\) from \(x=1\) to \(x=4\).
Problem 12: A rate of change of zero
Find the average rate of change of \(f(x)=x^{2}-2x\) from \(x=-1\) to \(x=3\). Explain why the answer can be zero even though the function is not constant.
Problem 13: Average rate of change from a graph
Use the points on the graph to find the average rate of change from \(x=0\) to \(x=2\), from \(x=2\) to \(x=5\), and from \(x=1\) to \(x=3\).
Problem 14: Rate of change of an exponential
Find the average rate of change of \(f(x)=2^{x}\) from \(x=1\) to \(x=3\) and from \(x=3\) to \(x=5\). What do the results show?
Problem 15: Equation versus table
Function \(f\) is \(f(x)=3x-2\). Function \(g\) is given by the table. Which function has the greater rate of change? Which has the greater \(y\)-intercept? Which has the greater value at \(x=10\)?
Problem 16: Equation versus graph
Function \(f\) is \(f(x)=-x^{2}+9\). Function \(g\) is shown in the graph. Compare the two: which has the greater maximum value, which has the greater \(y\)-intercept, and what are the \(x\)-intercepts of each?
Problem 17: Ball height features
A ball is hit upward. Its height in feet after \(t\) seconds is \(h(t)=-16t^{2}+64t+5\). Find the \(y\)-intercept and explain it. Find when the ball is highest and how high it gets. Find the average rate of change from \(t=0\) to \(t=2\).
Problem 18: Draining a tank
Water drains from a tank so that \(V(t)=120-8t\) gives the gallons left after \(t\) minutes. Find the intercepts and explain each. What is the rate of change, and what does it mean? State a reasonable domain.
Problem 19: Tide chart
The graph shows the water level at a dock, in feet above or below the dock level, \(t\) hours after midnight. When is the water above the dock? When is the level increasing? What are the relative maximum and minimum? Find the average rate of change from \(t=3\) to \(t=9\).
Problem 20: Road trip speeds
The table shows how many miles the Garcia family had driven \(t\) hours after leaving home. Find the average speed from \(t=0\) to \(t=4\), from \(t=1\) to \(t=3\), and from \(t=2\) to \(t=4\). During which of these three intervals was the average speed lowest?
Problem 21: Gym memberships
Gym A charges \(C_A(m)=20m+50\) dollars for \(m\) months. Gym B costs \(\$45\), \(\$90\), \(\$135\), \(\$180\) after 1, 2, 3, 4 months. Compare the rates of change. Find the cost of each after 1, 2 and 6 months. For which numbers of months is Gym A cheaper?
Problem 22: Exponential versus linear growth
Let \(F(x)=2^{x}\) and \(G(x)=10x\). Complete the comparison: find the average rate of change of each function from \(x=0\) to \(x=4\) and from \(x=0\) to \(x=6\). Which function has the greater rate on each interval?
See every chapter on the Algebra 1 math page.