Quadratic Function

Algebra 1

Quadratic Function

A quadratic function has the form \(f(x) = ax^2 + bx + c\) and always graphs as a parabola — a smooth U (or upside-down U). Its shape is set by a few features: which way it opens, its vertex, and where it crosses the axes. We’ll map them all, with a solver and a worksheet maker a tap away.

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Illustration of students learning Quadratic Function

A quadratic function is any rule of the form \(f(x) = ax^2 + bx + c\) (with \(a \ne 0\)). What makes it special is its graph: always a parabola, a graceful U-shaped curve. Understand a parabola’s handful of features — direction, vertex, axis of symmetry, intercepts — and you understand the whole function.

In short: a quadratic function is \(f(x) = ax^2 + bx + c\); it graphs as a parabola that opens up if \(a > 0\) and down if \(a < 0\), with its turning point (vertex) at \(x = -\tfrac{b}{2a}\).

The big idea

The Anatomy of a Parabola

The squared term is what bends the graph into a curve instead of a line. A few features describe any parabola completely:

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

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

Axis of symmetryThe vertical line through the vertex.

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

Tutor tip: The constant \(c\) is the y-intercept — no work needed, since \(f(0) = c\). The vertex is where the function reaches its minimum (opens up) or maximum (opens down).

A parabola, mapped

\(f(x) = x^2 – 6x + 5\)

It opens up (\(a = 1\)). Vertex: \(x = -\tfrac{-6}{2} = 3\), \(f(3) = -4\), so \((3,-4)\). It crosses the x-axis at \(1\) and \(5\) and the y-axis at \((0,5)\).

⚡ Analyze a quadratic

Worked Examples

Each parabola below is plotted from its equation, with the vertex and roots marked.

Example A — Opens up

Describe \(f(x) = x^2 – 6x + 5\).

\(a = 1 > 0\), so it opens up (a U).
Vertex: \(x = -\tfrac{-6}{2} = 3\), \(f(3) = -4\) → \((3,-4)\), the minimum.
Roots at \(x = 1, 5\); y-intercept \((0,5)\).

Answer: U with minimum \(-4\)

Example B — Opens down

Describe \(f(x) = -x^2 + 4\).

\(a = -1 < 0\), so it opens down (a frown).
Vertex \((0,4)\) is the maximum.
Roots at \(x = \pm 2\).

Answer: frown peaking at 4

Example C — Find the vertex

Find the vertex of \(f(x) = x^2 + 2x – 3\).

\(x = -\tfrac{b}{2a} = -\tfrac{2}{2} = -1\).
Substitute: \(f(-1) = 1 – 2 – 3 = -4\).
Vertex: \((-1, -4)\).

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

Example D — Read the y-intercept

What’s the y-intercept of \(f(x) = 2x^2 – 5x + 7\)?

The y-intercept is \(f(0)\).
\(f(0) = 2(0) – 5(0) + 7 = 7\) — just the constant.
Point: \((0, 7)\). This parabola opens up and never crosses the x-axis.

Answer: \((0, 7)\)

Quadratics in the Wild

Parabolas describe anything that rises and then falls — or the reverse. A thrown ball’s height, a company’s profit as price changes, the cable of a suspension bridge, the dish of a satellite antenna. The vertex marks the peak height, the maximum profit, or the lowest point of the cable, which is exactly why this one shape matters so much.

Slip-Ups That Cost Easy Points

Missing the sign of \(a\). A negative \(a\) opens the parabola downward.
Vertex formula slip. It’s \(x = -\tfrac{b}{2a}\); don’t forget the negative, then plug back in for \(y\).
Confusing vertex and intercepts. The vertex is the turning point; the intercepts are where it crosses the axes — different points.
Forgetting \(a \ne 0\). If \(a = 0\) it’s linear, not quadratic — there’s no parabola.

Your Turn

Give the vertex, the y-intercept, and which way it opens. Reveal to check.

\(f(x) = x^2 – 4\)
\(f(x) = x^2 – 2x – 8\)
\(f(x) = 2x^2\)

Show answers

\(\color{blue}{\text{vertex }(0,-4),\ \text{y-int }(0,-4),\ \text{up}}\)
\(\color{blue}{\text{vertex }(1,-9),\ \text{y-int }(0,-8),\ \text{up}}\)
\(\color{blue}{\text{vertex }(0,0),\ \text{y-int }(0,0),\ \text{up (narrow)}}\)

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Frequently Asked Questions

What makes a function quadratic?

It has a squared variable term and the form \(f(x) = ax^2 + bx + c\) with \(a \ne 0\). Its graph is always a parabola.

How do I find the vertex?

Use \(x = -\tfrac{b}{2a}\) for the vertex’s x-coordinate, then substitute it back to get \(y\). The vertex is the curve’s minimum (if it opens up) or maximum (if it opens down).

Which way does the parabola open?

Up if \(a > 0\), down if \(a < 0\). A larger \(|a|\) makes it narrower; a smaller \(|a|\) makes it wider.

What’s the difference between a quadratic function and a quadratic equation?

A quadratic function \(f(x) = ax^2 + bx + c\) is a rule you graph; a quadratic equation sets it equal to a value (usually 0) and asks for the x-values that solve it.

Continue Your Study

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

Up nextHow to Solve a Quadratic EquationContinue with Quadratic Functions and EquationsOpen lesson →Up nextGraphing Quadratic FunctionsContinue with Quadratic Functions and EquationsOpen lesson →Up nextSolving a Quadratic Equation by Completing the SquareContinue with Quadratic Functions and EquationsOpen lesson →

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Quadratic Function