Graphing Parabolas
Find the vertex, axis of symmetry and intercepts, then sketch.
About this topic
To graph a parabola, first find the axis of symmetry x = −b/(2a), then the vertex by substituting that x back in. Find the y-intercept c, and mirror it across the axis, plus the x-intercepts if they exist. A handful of points gives a smooth curve.
The vertex is the minimum point when the parabola opens up and the maximum when it opens down. For y = x² − 6x + 5, the axis is x = 3, the vertex is (3, −4), the y-intercept is (0, 5) with mirror point (6, 5), and the x-intercepts come from factoring: (x − 1)(x − 5), so x = 1 and 5. Plotting these five points gives an accurate curve.
This process is used in physics to find the peak height and landing point of a projectile, and in business to find the price that maximizes revenue. Graphing calculators show the same features, but knowing how to find them by hand lets you check the result.
Draw a smooth curve, not straight segments, and do not flatten the bottom into a point. Be careful with the sign in −b/(2a): if b is negative, the axis is positive. Symmetry saves effort, so use it to double your points.
Key ideas
- Axis of symmetry is x = −b/2a
- Plug into the function for the vertex y
- Mirror points across the axis
Quick check
Use cases: where Graphing Parabolas helps
- Homework: review the key idea behind Graphing Parabolas before you start a problem set.
- Exam revision: use the quick check to test whether you can recall Graphing Parabolas without notes.
- Visual learning: play with the interactive lab to see how Graphing Parabolas changes when you adjust its inputs.
- Teaching a friend: explain Graphing Parabolas in your own words, then compare your version with the key ideas.
Number and algebra skills are used in science, engineering, budgeting and computing.
Frequently asked questions
What is Graphing Parabolas?
Find the vertex, axis of symmetry and intercepts, then sketch.
What will I learn in Graphing Parabolas?
Axis of symmetry is x = −b/2a. Plug into the function for the vertex y. Mirror points across the axis.
Is Graphing Parabolas part of Mathematics?
Yes. It is one of 200 Mathematics topics on LearnViz HighSchoolEdu, with 3 key ideas, a live animation and a quick check.
Does the Graphing Parabolas lab work on a phone or tablet?
Yes. The animation resizes to fit your screen and supports touch, so you can drag, tap and slide with your finger.
How should I study Graphing Parabolas?
Read the summary and key ideas first, play with the interactive lab to see the idea in motion, then answer the quick check. If you get a question wrong, go back to the key idea it tests.
Do I need an account, and is my progress saved?
No account is needed. Marking a topic as learned is saved in your own browser, so it stays on that device and is not sent to a server.