Quadratic Function

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.

Tutor-style math help

Quadratic Function: what to notice and how to work it

Quadratics skill
Quadratic topics connect an equation, a parabola, roots, and a turning point. Read the form first because each form reveals a different feature.

What to notice first

Standard form helps with formulas, factored form shows roots, and vertex form shows the turning point. A good solution starts by using the form you have.

Common student mistake

Do not assume every quadratic has two real x-intercepts. The discriminant tells whether the real graph crosses the x-axis twice, once, or not at all.

Key formulas and cues

\(ax^2+bx+c=0\)
\(x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}\)
\(x=-\frac{b}{2a}\)
\(y=a(x-h)^2+k\)
vertex axis

A reliable path

  1. Read the formFactored, standard, and vertex forms reveal different features.
  2. Choose the methodFactor when friendly, complete the square for structure, or use the formula when needed.
  3. Connect to the graphRoots are x-intercepts and the vertex is the minimum or maximum point.

Worked examples

Factor and solve

Example: \(x^2-7x+12=0\)
  1. Factor into (x – 3)(x – 4).
  2. Set each factor equal to zero.
  3. Solve both small equations.
Answer: \(x=3\) or \(x=4\)

Find the axis

Example: \(y=2x^2-8x+5\)
  1. Use x = -b/(2a).
  2. Here a = 2 and b = -8.
  3. Compute 8/4.
Answer: \(x=2\)
Try one before moving on
Try: Find the axis of symmetry of \(y=x^2-6x+2\).
Answer: \(x=3\).
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
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
vertex (3, -4)

Worked Examples

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

Example A — Opens up

Describe \(f(x) = x^2 – 6x + 5\).

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

Answer: U with minimum \(-4\)

vertex (3, -4)

Example B — Opens down

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

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

Answer: frown peaking at 4

vertex (0, 4)

Example C — Find the vertex

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

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

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

vertex (-1, -4)

Example D — Read the y-intercept

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

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

Answer: \((0, 7)\)

vertex (5/4, 31/8)

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.

  1. \(f(x) = x^2 – 4\)
  2. \(f(x) = x^2 – 2x – 8\)
  3. \(f(x) = 2x^2\)
Show answers
  1. \(\color{blue}{\text{vertex }(0,-4),\ \text{y-int }(0,-4),\ \text{up}}\)
  2. \(\color{blue}{\text{vertex }(1,-9),\ \text{y-int }(0,-8),\ \text{up}}\)
  3. \(\color{blue}{\text{vertex }(0,0),\ \text{y-int }(0,0),\ \text{up (narrow)}}\)
Keep practicing

Make Your Own Quadratics Worksheet

Generate fresh quadratic-function problems with a full answer key — print or save as a PDF.

New problems every click — never the same sheet twice
Step-by-step answer key so you can self-check
📈

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.

Related Topics

Continue Your Study

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

Related to This Article

What people say about "Quadratic Function - Effortless Math"?

No one replied yet.

Leave a Reply

X
51% OFF

Limited time only!

Save Over 51%

Take It Now!

SAVE $55

It was $109.99 now it is $54.99

The Ultimate Algebra Bundle 2026: From Pre-Algebra to Algebra II