Quadratic Function
Quick answer: A quadratic function has the form f(x) = ax² + bx + c, where a ≠ 0. Its graph is a U-shaped parabola that opens up when a > 0 and down when a < 0. The vertex is at x = −b/(2a). Example: f(x) = x² − 6x + 5 opens up, has vertex (3, −4) and crosses the x-axis at 1 and 5.
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 quadraticStandard, Vertex and Factored Form
The same quadratic function can be written in three ways. Each one makes a different feature easy to read, so it helps to know all three.
| Form | General pattern | What it shows right away | Example: same parabola |
|---|---|---|---|
| Standard | f(x) = ax² + bx + c | Direction (sign of a) and y-intercept (0, c) | f(x) = x² − 6x + 5 |
| Vertex | f(x) = a(x − h)² + k | Vertex (h, k) and axis x = h | f(x) = (x − 3)² − 4 |
| Factored | f(x) = a(x − r)(x − s) | x-intercepts r and s | f(x) = (x − 1)(x − 5) |
Check that they match. Expand the vertex form: (x − 3)² − 4 = x² − 6x + 9 − 4 = x² − 6x + 5. Expand the factored form: (x − 1)(x − 5) = x² − 5x − x + 5 = x² − 6x + 5. All three are the same function.
Standard to vertex form. Complete the square. For x² − 6x + 5, take half of −6, which is −3, and square it to get 9. Add and subtract 9: x² − 6x + 9 − 9 + 5 = (x − 3)² − 4.
Standard to factored form. Look for two numbers that multiply to c and add to b. For x² − 6x + 5, the numbers −1 and −5 work, so f(x) = (x − 1)(x − 5). Not every quadratic factors nicely over the integers; when it does not, use the quadratic formula to find the roots.
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: \((0, 7)\)
The Discriminant: How Many x-Intercepts?
Before you try to solve or graph, the discriminant tells you how many times the parabola crosses the x-axis. It is the part of the quadratic formula under the square root:
D = b² − 4ac
| Value of D | Real roots | What the graph does | Example |
|---|---|---|---|
| D > 0 | Two | Crosses the x-axis twice | x² − 6x + 5: D = 36 − 20 = 16 |
| D = 0 | One (repeated) | Touches the x-axis at the vertex | x² − 4x + 4: D = 16 − 16 = 0, touching at (2, 0) |
| D < 0 | None | Stays entirely above or below the x-axis | 2x² − 5x + 7: D = 25 − 56 = −31 |
The last row explains Example D above. Its parabola opens up and its lowest point, (5/4, 31/8), is above the x-axis, so it never crosses. The negative discriminant tells you that without drawing anything.
A perfect-square discriminant, such as 16, also tells you the quadratic factors over the integers. That is a quick hint that factoring will be faster than the formula.
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.
A Real-World Problem, Fully Worked
A ball is thrown upward from the top of a 48-foot building. Its height in feet after t seconds is h(t) = −16t² + 32t + 48. How high does it go, and when does it hit the ground?
- Direction. a = −16 is negative, so the parabola opens down and the vertex is the highest point.
- Time of the peak. t = −b/(2a) = −32/(2 × −16) = 1 second.
- Maximum height. h(1) = −16 + 32 + 48 = 64 feet.
- Hitting the ground. Set h(t) = 0: −16t² + 32t + 48 = 0. Divide by −16 to get t² − 2t − 3 = 0, which factors as (t − 3)(t + 1) = 0.
- Pick the answer that makes sense. t = 3 or t = −1. Time cannot be negative here, so the ball lands after 3 seconds.
Notice how each feature has a meaning: the y-intercept 48 is the starting height, the vertex (1, 64) is the peak, and the positive root 3 is the landing time. Reading a quadratic this way is the main skill tested in word problems.
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.
Practice Quiz: 10 Questions
Answer each question and press Submit to see your result right away.
Answer each question and press Submit to see your result right away.
Frequently Asked Questions
What is a quadratic function in math?
A quadratic function is a function of the form f(x) = ax² + bx + c, where a, b and c are numbers and a ≠ 0. Its highest power of x is 2, and its graph is a U-shaped curve called a parabola. If a is positive the parabola opens up and has a minimum; if a is negative it opens down and has a maximum.
What is a quadratic function in vertex form?
Vertex form is y = a(x − h)² + k, where (h, k) is the vertex of the parabola and x = h is its axis of symmetry. For example, y = 2(x − 1)² + 3 has vertex (1, 3) and opens up because a = 2 is positive, so its minimum value is 3. Watch the sign: y = (x + 4)² has h = −4, not 4.
How do you graph quadratic functions in standard form?
Find the axis of symmetry with x = −b/(2a), plug it in to get the vertex, then plot the y-intercept and a few points on each side. For y = x² − 4x + 3, the axis is x = 4/2 = 2, and y = 4 − 8 + 3 = −1, so the vertex is (2, −1). The y-intercept is 3, and the x-intercepts are 1 and 3.
How do you solve quadratic functions?
Set the function equal to zero and solve by factoring, completing the square or the quadratic formula, x = (−b ± √(b² − 4ac))/(2a). For x² + 2x − 15 = 0, find two numbers that multiply to −15 and add to 2, which are 5 and −3. So (x + 5)(x − 3) = 0, and x = −5 or x = 3. These are the graph’s x-intercepts.
What is the domain and range of a quadratic function?
The domain of any quadratic function is all real numbers. The range depends on the vertex (h, k) and the direction it opens: y ≥ k if a > 0, and y ≤ k if a < 0. For y = −(x − 2)² + 5, the parabola opens down with vertex (2, 5), so the range is y ≤ 5.
Is a quadratic function even or odd?
A quadratic function is even only when b = 0, as in f(x) = x² + 1, because then f(−x) = f(x) and the graph is symmetric about the y-axis. Otherwise it is neither. For f(x) = x² + 2x, f(1) = 3 and f(−1) = −1, which are not equal and not opposites. A quadratic function is never odd.
How do you know if a function is quadratic? In an equation, the highest power of x is 2 and the x² term does not cancel out. In a table with evenly spaced x-values, the first differences of y change but the second differences stay the same. On a graph, the shape is a parabola. For example, y = 3x² − x is quadratic, but y = x³ + 2 is not.
What is the difference between a quadratic function and a quadratic equation? A quadratic function is a rule, f(x) = ax² + bx + c, that gives an output for every x and graphs as a parabola. A quadratic equation sets the expression equal to a value, usually 0, as in x² − 6x + 5 = 0. Its solutions, 1 and 5, are the x-intercepts of the function.
How many x-intercepts can a quadratic function have? Two, one or none. Use the discriminant b² − 4ac. If it is positive, the parabola crosses the x-axis twice. If it is zero, the vertex sits on the x-axis. If it is negative, the graph never touches the x-axis. A quadratic function always has exactly one y-intercept, at (0, c).
What does the value of a do to a parabola? The sign of a sets the direction: positive opens up, negative opens down. The size of a sets the width. When the absolute value of a is greater than 1, the parabola is narrower than y = x². When it is between 0 and 1, it is wider. For example, y = 3x² is narrow and y = 0.5x² is wide.
Related Topics
Continue Your Study
Ready for the next step? Pick up right where this lesson leaves off:
Related to This Article
More math articles
- The Ultimate 7th Grade NDSA Math Course (+FREE Worksheets)
- Dividing Dollars: How to Navigate Money Division in Word Problems
- Evaluating Variables and Expressions
- 3rd Grade WY-TOPP Math Worksheets: FREE & Printable
- How to Read Scientific Diagrams, Tables, and Graphs
- House and Senate voting
- Select and special committees
- 15 Geniuses Who Changed the World of Mathematics Forever
- How to Multiply Monomials? (+FREE Worksheet!)
- Resource Mobilization
What people say about "Quadratic Function: Forms, Graph, Vertex and Examples"?
No one replied yet.