Graphs and Solutions Worksheet with Answers – Algebra 1 (Free PDF)

Problem 1 Test Two Points on a Line ★★★

Decide whether each point is on the graph of y = 2x + 5: (3, 11) and (−1, 4).

Problem 2 Test Three Points on a Parabola ★★★

Decide which of the points (4, 10), (−3, 3) and (2, −1) are on the graph of y = x² − 6.

Problem 3 Complete the Solution Pair ★★★

Find the missing coordinate so that each point is on the graph.

1. (4, ?) on y = 3x − 7
2. (−3, ?) on y = x² + 2x

Problem 4 Growth y-Intercept and a Value ★★★

For y = 5(2)^x, give the y-intercept and the value at x = 3. Does the graph rise or fall as x increases?

Problem 5 Decay y-Intercept and a Value ★★★

For y = 80(12)^x, give the y-intercept, the value at x = 3 and the horizontal asymptote.

Problem 6 Negative Inputs ★★★

For y = 6(3)^x, find y when x = −1 and when x = −2.

Problem 7 Find Intersections from a Graph ★★★

The graph shows y = x² − 4 and y = x − 2. Estimate the intersection points, then confirm them algebraically.

Problem 8 Substitution with y = x² ★★★

Solve the system y = x² and y = 2x + 8.

Problem 9 System with a Missing Constant Term ★★★

Solve the system y = x² − x − 6 and y = 2x − 6.

Problem 10 Exactly One Solution ★★★

Solve the system y = x² − 6x + 10 and y = 2x − 6. What does the number of solutions tell you about the graphs?

Problem 11 No Solution ★★★

Show that the system y = x² + 3 and y = 2x has no real solution.

Problem 12 Tangent Condition ★★★

For what value of c does the line y = 2x + c touch the parabola y = x² at exactly one point? Find that point.

Problem 13 Read an Exponential Graph ★★★

The graph shows an exponential function. Give its y-intercept, its horizontal asymptote and its end behavior as x → ∞ and as x → −∞. Write its equation.

Problem 14 Rewrite with the Power Rule ★★★

Rewrite 100(1.08)^t, where t is in years, so that the exponent counts months, n = 12t. Find the monthly growth factor to four decimal places.

Problem 15 Ball Over a Ramp ★★★

A ball follows the path h = −x² + 6x, and a ramp is the line h = x + 4 (both in feet). Find where the ball path meets the ramp.

Problem 16 Solve and Check Both Equations ★★★

Solve y = x² − 2x − 8 and y = −x + 4. Check each solution in both equations.

Problem 17 Monthly Rate for a Savings Account ★★★

An account grows by 6% each year: A = 1500(1.06)^t. Rewrite it with an exponent that counts months, find the monthly growth rate (to the nearest hundredth of a percent) and find the balance after 18 months.

Problem 18 Half-Life as Yearly Decay ★★★

A radioactive sample has mass M = 200(12) ^t5 grams after t years. Rewrite it as 200b^t, find the yearly decay rate to the nearest tenth of a percent and find the mass after 10 years.

Problem 19 Intercepts and End Behavior ★★★

The graph shows f(x) = 3(2)^x − 6. Find the y-intercept and the x-intercept, name the horizontal asymptote and describe the end behavior as x → ∞ and x → −∞.

Problem 20 Irrational Intersection Points ★★★

Find the intersection points of y = x² and y = 2x + 1. Give exact coordinates for the x-values and decimals to the nearest hundredth.