Quadratic Function
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.
Quadratic Function: what to notice and how to work it
What to notice first
Common student mistake
Key formulas and cues
A reliable path
- Read the formFactored, standard, and vertex forms reveal different features.
- Choose the methodFactor when friendly, complete the square for structure, or use the formula when needed.
- Connect to the graphRoots are x-intercepts and the vertex is the minimum or maximum point.
Worked examples
Factor and solve
- Factor into (x – 3)(x – 4).
- Set each factor equal to zero.
- Solve both small equations.
Find the axis
- Use x = -b/(2a).
- Here a = 2 and b = -8.
- Compute 8/4.
Try one before moving on
Quadratic Function: pop-up practice
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 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:
\(f(x) = x^2 – 6x + 5\)
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 quadraticWorked 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
- \(\color{blue}{\text{vertex }(0,-4),\ \text{y-int }(0,-4),\ \text{up}}\)
- \(\color{blue}{\text{vertex }(1,-9),\ \text{y-int }(0,-8),\ \text{up}}\)
- \(\color{blue}{\text{vertex }(0,0),\ \text{y-int }(0,0),\ \text{up (narrow)}}\)
Make Your Own Quadratics Worksheet
Generate fresh quadratic-function problems with a full answer key — print or save as a PDF.
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
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