How to Graph Quadratic Functions? (+FREE Worksheet!)

Algebra 1

How to Graph Quadratic Functions

A quadratic function graphs as a parabola — a smooth U (or upside-down U). Find its vertex, its line of symmetry, and where it crosses the axes, and you can sketch any of them. We’ll work through it with verified graphs, a solver, and a worksheet maker a tap away.

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Illustration of students learning How to Graph Quadratic Functions

To graph a quadratic function like \(y = ax^2 + bx + c\), you sketch a parabola — a smooth U-shaped curve. It might open up like a smile or down like a frown, sit high or low, stretch wide or narrow. But every parabola has the same handful of landmarks, and once you can find them, you can sketch any quadratic with confidence. Let’s learn those landmarks one at a time.

The big idea

The Parts of a Parabola

To graph \(y = ax^2 + bx + c\), you only need a few features:

Direction\(a>0\) opens up; \(a<0\) opens down.

VertexThe turning point, at \(x=-\tfrac{b}{2a}\).

Axis of symmetryThe vertical line \(x=-\tfrac{b}{2a}\) through the vertex.

Intercepts\(y\)-intercept at \((0,c)\); \(x\)-intercepts where \(y=0\).

Vertex form\(y=a(x-h)^2+k\) shows the vertex \((h,k)\) directly.

How to graph a quadratic (4 steps):

Check the sign of \(a\) to see which way it opens.
Find the vertex with \(x=-\tfrac{b}{2a}\), then compute \(y\).
Plot the \(y\)-intercept \((0,c)\) and any \(x\)-intercepts.
Use symmetry to mirror points across the axis, then draw the curve.

Worked on the grid

Graphing \(y = x^2 – 6x + 5\)

It opens up (\(a=1>0\)). Vertex: \(x=-\tfrac{-6}{2}=3\), and \(y=3^2-6(3)+5=-4\), so \((3,-4)\). It crosses the \(x\)-axis at \(1\) and \(5\), and the \(y\)-axis at \((0,5)\). Connect with a smooth U.

⚡ Graph a quadratic

Worked Examples

Find the landmarks, then connect — every parabola below is plotted straight from its equation.

Example A — Opens up

Graph \(y = x^2 – 6x + 5\).

\(a = 1 > 0\): opens up. Vertex \(x = -\tfrac{-6}{2} = 3\), \(y = -4\) → \((3,-4)\).
y-intercept \((0,5)\); factor \((x-1)(x-5)\) → roots 1 and 5.
Connect with a smooth U.

Answer: U, vertex \((3,-4)\)

Example B — Find the vertex

Graph \(y = x^2 + 2x – 3\).

Vertex: \(x = -\tfrac{2}{2} = -1\), \(y = -4\) → \((-1,-4)\).
Factor \((x+3)(x-1)\) → roots \(-3\) and 1.
Draw the U through those points.

Answer: vertex \((-1,-4)\)

Example C — Opens down

Graph \(y = -x^2 + 4\).

\(a = -1 < 0\): opens down. Vertex \((0,4)\), the maximum.
Set \(y = 0\): \(x = \pm 2\).
Draw the frown peaking at \((0,4)\).

Answer: frown, vertex \((0,4)\)

Example D — Use symmetry

For \(y = x^2 – 6x + 5\), confirm a mirrored root.

Axis of symmetry is \(x = 3\).
The root \((1,0)\) is 2 units left of the axis.
Its twin is 2 units right: \((5,0)\) — symmetry doubles your points for free.

Answer: mirror root \((5,0)\)

Example E — A fractional vertex

Graph \(y = x^2 – 3x + 1\).

Vertex: \(x = -\tfrac{-3}{2} = \tfrac32\).
Substitute: \(y = \left(\tfrac32\right)^2 – 3\left(\tfrac32\right) + 1 = -\tfrac54\).
Vertex \(\left(\tfrac32, -\tfrac54\right)\) — vertices need not be whole numbers.

Answer: vertex \(\left(\tfrac32, -\tfrac54\right)\)

Parabolas in the Wild

Throw a ball and its height over time traces a parabola. A quadratic like \(h = -16t^2 + 32t\) (height in feet, time in seconds) opens down because gravity pulls it back; its vertex is the highest point of the throw, and its \(x\)-intercepts are when the ball leaves and lands. Satellite dishes, headlight reflectors, and suspension-bridge cables are parabolas too — the shape shows up wherever something is focused or falls.

Slip-Ups That Cost Easy Points

Forgetting the sign of \(a\). A negative \(a\) opens downward. If your “U” doesn’t match the equation’s sign, recheck.
Vertex formula slip. It’s \(x=-\tfrac{b}{2a}\) — don’t forget the negative, and remember to plug that \(x\) back in to get \(y\).
Stopping at the vertex. One point isn’t a parabola. Plot the intercepts and use symmetry for a believable curve.
Reading the \(y\)-intercept wrong. It’s always \((0, c)\) — the constant term — found by setting \(x=0\).

Your Turn: Find Vertex, Roots, and Direction

For each quadratic, give the vertex, the \(x\)-intercepts (roots), and whether it opens up or down. Reveal to check.

\(y = x^2 – 4x + 3\)
\(y = x^2 – 2x – 8\)
\(y = -x^2 + 6x – 5\)
\(y = x^2 – 4x + 4\)

Show answers

\(\color{blue}{\text{vertex }(2,-1),\ \text{roots }1,3,\ \text{opens up}}\)
\(\color{blue}{\text{vertex }(1,-9),\ \text{roots }-2,4,\ \text{opens up}}\)
\(\color{blue}{\text{vertex }(3,4),\ \text{roots }1,5,\ \text{opens down}}\)
\(\color{blue}{\text{vertex }(2,0),\ \text{one (double) root }x=2,\ \text{opens up}}\)

Keep practicing

Make Your Own Quadratics Worksheet

Generate fresh quadratic functions to graph, with a full answer key — print or save as a PDF.

Fresh parabolas every click — never the same set twice

Worked answer key for every problem

📄 Open the Worksheet Maker 📇 Formula Flashcards

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Frequently Asked Questions

How do I find the vertex of a parabola?

Use \(x=-\tfrac{b}{2a}\) to get the vertex’s \(x\)-coordinate, then substitute that value back into the equation to find \(y\). That point is the highest or lowest point of the parabola.

How do I know if a parabola opens up or down?

Look at \(a\), the coefficient of \(x^2\). If \(a>0\) it opens up (a smile with a minimum); if \(a<0\) it opens down (a frown with a maximum).

What are the \(x\)-intercepts and how do I find them?

They’re where the parabola crosses the \(x\)-axis, found by setting \(y=0\) and solving the quadratic — by factoring, the quadratic formula, or a solver. A parabola can have two, one, or zero \(x\)-intercepts — for example, \(y=x^2+1\) never touches the \(x\)-axis, so it has zero.

What is the axis of symmetry?

It’s the vertical line \(x=-\tfrac{b}{2a}\) that splits the parabola into mirror halves and passes through the vertex. Any point on one side has a twin the same distance on the other side.

Continue Your Study

Ready for the next step? Pick up right where this lesson leaves off:

Up nextNumber of Solutions in a System of EquationsContinue with Inequalities and Systems of EquationsOpen lesson →Up nextSystems of Linear InequalitiesContinue with Inequalities and Systems of EquationsOpen lesson →Up nextHow to Solve One-Step InequalitiesContinue with Inequalities and Systems of EquationsOpen lesson →

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