Parabolas: How to Graph a Quadratic Function

Drop a ball, aim a satellite dish, or trace the cable on a suspension bridge and you get the same curve. A parabola is the graph of a quadratic — any function whose highest power of is 2 — and it is the second shape most algebra courses teach after the straight line.

What Makes a Curve a Parabola

A parabola is the graph of

That matters. If were zero the term would vanish and you would be left with — a straight line, not a curve.

Two features define every parabola:

• One turning point, the vertex, where the curve stops falling and starts rising (or the reverse).
• A vertical mirror line through that vertex, the axis of symmetry. Fold the page along it and the two halves match exactly.

That symmetry is not decoration — it is the single most useful property when sketching, because it means every point you plot on one side gives you a second point free.

What the Coefficient a Controls

The number in front of does two jobs at once.

Its sign sets the direction. Positive opens the curve upward, so the vertex is a minimum. Negative opens it downward, so the vertex is a maximum.

Its size sets the width. A large squeezes the curve narrow; a small spreads it wide.

A quick sanity check: at one unit right of the vertex, the curve has risen exactly . At two units right it has risen . That “1, 4, 9 times ” pattern is worth memorising — it is how you place points without a calculator.

The Two Forms

Standard form is . Its gift is the -intercept: setting leaves , so you can read it off with no work at all.

Vertex form is

and its gift is the vertex, sitting right there as . As with the equation of a circle, the bracket subtracts the coordinate, so you negate what you see: means .

Read that way, vertex form is just picked up and moved: right by , up by , and stretched by . Converting standard form into vertex form is exactly the completing the square procedure used on circles.

Finding the Intercepts

The -intercept is . One point, free.

The -intercepts are wherever , so they are the solutions of — which is to say, they are the roots of the quadratic equation. Solve by factoring when you can and by the quadratic formula when you cannot.

How many there are is decided before you solve anything, by the discriminant :

A parabola with no -intercepts is not an error. sits a full unit above the axis and never reaches it, which is a perfectly good answer.

The Five-Point Method

Tables of values waste effort, because symmetry means half the rows are predictable. This is faster and more accurate:

1. Find the vertex. Either read it from vertex form, or use and substitute back — the method covered in vertex and axis of symmetry.
2. Draw the axis of symmetry as a dashed vertical line through it.
3. Plot two points on one side. One and two units across is usually enough.
4. Mirror them to the other side at the same heights.
5. Join the five points with a smooth curve — never straight segments, and never a sharp corner at the vertex.

Worked Example A: Sketch From Vertex Form

Sketch .

The vertex is and , so it opens upward and is narrower than .

One unit right of the vertex the curve has risen , giving . Two units right it has risen , giving . Mirroring across gives and .

Five points, and the -intercepts came out of the working for free.

Worked Example B: Sketch From Standard Form

Sketch .

The -intercept is .

The axis of symmetry is , and , so the vertex is .

For the -intercepts, , giving and .

Plot , , , , and the mirror of the -intercept at .

Worked Example C: A Downward Parabola

Describe the graph of .

is negative, so it opens downward and the vertex is a maximum. The bracket means , so the vertex is .

Because the curve is wider than . It has a maximum value of 6, reached at .

Worked Example D: Counting the Intercepts First

How many -intercepts does have?

Negative, so none. Since the parabola opens upward, so it sits entirely above the -axis.

Worked Example E: Building the Equation From a Graph

A parabola has vertex and passes through . Find its equation.

Start in vertex form: . Substituting the known point,

So . One extra point is always enough to pin down once you know the vertex.

Common Mistakes to Avoid

• Getting the vertex signs backwards. means , not 3.
• Assuming every parabola crosses the -axis. Plenty do not, and “no real roots” is a complete answer.
• Drawing a sharp point at the vertex. A parabola turns smoothly; only absolute-value graphs have a corner.
• Reading as the width alone. Its sign carries the direction too.
• Plotting points blindly on both sides. Symmetry halves the work.
• Forgetting when the curve is stretched. At one unit from the vertex the rise is , not 1.
• Confusing the axis of symmetry with the vertex. The axis is a line, ; the vertex is a point.

Where Parabolas Lead Next

• Maximum and minimum problems. The vertex answers every “largest area” and “greatest height” question — see vertex and axis of symmetry.
• Conic sections. A parabola is the conic you get by slicing a cone parallel to its side; circles, ellipses and hyperbolas are the rest of the family.
• Projectile motion. Under constant gravity, height against time is quadratic.
• Quadratic inequalities. Knowing where the curve sits above or below the axis solves them by inspection.
• Reflector design. A parabolic dish focuses every incoming parallel ray onto one point, which is why satellite dishes and headlight reflectors have this cross-section.

Practice Problems

Work each one before opening the answer.

Problem 1. Which way does open, and is its vertex a maximum or a minimum?

Problem 2. State the vertex of .

Problem 3. State the vertex of .

Problem 4. What is the -intercept of ?

Problem 5. Find the -intercepts of .

Problem 6. How many -intercepts does have?

Problem 7. How many -intercepts does have?

Problem 8. Is narrower or wider than ?

Problem 9. Write in standard form.

Problem 10. A parabola has vertex and passes through . Find its equation.

Problem 11. A parabola has vertex and passes through . Find its equation in vertex form.

Problem 12. Sketch using five points.

Quick Reference

Parabolas are the curve half of graphing equations, and their intercepts are the roots you find in quadratic equations. The next lesson, vertex and axis of symmetry, goes deeper on locating that turning point. To drill the algebra underneath, try the quadratic equations practice problems. More Algebra lessons are available.