Die Druckversion wird nicht mehr unterstützt und kann Darstellungsfehler aufweisen. Bitte aktualisiere deine Browser-Lesezeichen und verwende stattdessen die Standard-Druckfunktion des Browsers.

Graphs of Quadratic Functions

Introduction

This course is designed for learners in Grades 9–10. A quadratic function is a function that can be written in the form $f (x) = a x^{2} + b x + c$ , where $a \neq 0$ . Its graph is a parabola, a smooth U-shaped curve that may open upward or downward. In this course, you will learn how an equation controls the shape and position of its graph, how to identify key features, and how to move between equations, tables, graphs, and real-world situations.

Quadratic graphs are important in algebra because they connect symbolic rules with visual patterns. They also appear in simplified models of projectile motion, optimization, area problems, engineering design, and computer graphics.

By the end of this aiMOOC, you should be able to recognize a quadratic function, graph it accurately, interpret its vertex and intercepts, describe transformations, connect roots with x-intercepts, and use a quadratic model to reason about a situation.

Learning Goals

After working through the course, you can:

Quadratic function: Recognize a quadratic function in standard, vertex, and factored form.
Parabola: Explain how the leading coefficient affects opening direction and width.
Vertex: Find and interpret the vertex and axis of symmetry.
Intercept: Determine x-intercepts and the y-intercept when they exist.
Function transformation: Describe horizontal shifts, vertical shifts, reflections, and vertical stretches or compressions.
Quadratic equation: Connect roots of an equation with x-intercepts of the graph.
Domain and range: State the domain and determine the range from the vertex and opening direction.
Mathematical model: Use a quadratic function to model and interpret a real situation.

What Makes a Function Quadratic?

A single-variable quadratic function has degree two. In standard form,

$f (x) = a x^{2} + b x + c, a \neq 0$ .

The number $a$ is the leading coefficient. The number $b$ is the coefficient of the linear term, and $c$ is the constant term.

The graph of every real quadratic function is a parabola. If $a > 0$ , the parabola opens upward and has a minimum point. If $a < 0$ , it opens downward and has a maximum point.

Notice that the two curves in the image have the same vertex but opposite opening directions. Changing the sign of the leading coefficient reflects the graph across the x-axis.

The Parent Function

The simplest quadratic function is $f (x) = x^{2}$ . It is often called the parent function because many other quadratic graphs can be understood as transformations of this graph.

A quick table gives symmetric points:

x	-3	-2	-1	0	1	2	3
$y = x^{2}$	9	4	1	0	1	4	9

The equal y-values for opposite x-values show the symmetry of the parent parabola about the y-axis.

Key Features of a Parabola

When you graph a quadratic function, look for a small set of features that describe the whole curve.

Vertex and Axis of Symmetry

The vertex is the turning point of the parabola. It is the minimum point when the parabola opens upward and the maximum point when the parabola opens downward.

The axis of symmetry is the vertical line through the vertex. If the vertex is $(h, k)$ , then the axis of symmetry is $x = h$ .

For a quadratic in standard form $f (x) = a x^{2} + b x + c$ , the x-coordinate of the vertex is

$x = - \frac{b}{2 a}$ .

Substitute this x-value into the function to find the y-coordinate of the vertex.

Intercepts

The y-intercept occurs where $x = 0$ . In standard form, $f (0) = c$ , so the y-intercept is $(0, c)$ .

An x-intercept occurs where $f (x) = 0$ . The x-values are also called the zeros or roots of the corresponding quadratic equation. A parabola can cross the x-axis twice, touch it once, or miss it entirely.

Domain and Range

For a quadratic function with real coefficients, the domain is all real numbers because any real x-value can be substituted into the formula.

The range depends on the vertex and the opening direction. If the vertex is $(h, k)$ and the graph opens upward, then $y \geq k$ . If it opens downward, then $y \leq k$ .

Three Useful Forms of a Quadratic Function

The same quadratic function can often be written in different forms. Each form makes different graph features easier to see.

Standard Form

$f (x) = a x^{2} + b x + c$

Standard form makes the y-intercept easy to identify because it is $(0, c)$ . The leading coefficient $a$ tells you whether the graph opens upward or downward. The vertex can be found using $x = - \frac{b}{2 a}$ .

Example: $f (x) = x^{2} - 4 x + 3$ . Here $a = 1$ , $b = - 4$ , and $c = 3$ . The axis of symmetry is $x = 2$ , and the vertex is $(2, - 1)$ .

Vertex Form

$f (x) = a (x - h)^{2} + k$

Vertex form shows the vertex directly: $(h, k)$ . The axis of symmetry is $x = h$ .

Example: $f (x) = 2 (x - 3)^{2} - 5$ has vertex $(3, - 5)$ . It opens upward because $a = 2 > 0$ , and it is narrower than $y = x^{2}$ because $| a | > 1$ .

Factored Form

$f (x) = a (x - r_{1}) (x - r_{2})$

When the roots are real, factored form shows the x-intercepts directly: $x = r_{1}$ and $x = r_{2}$ .

Example: $f (x) = (x - 1) (x - 3)$ has x-intercepts at $(1, 0)$ and $(3, 0)$ . Its axis of symmetry lies halfway between the roots at $x = 2$ .

How to Graph a Quadratic Function

There is no single best graphing method for every equation. Choose a method that uses the form you are given.

Method A: Make a Table of Values

This method works for any quadratic function.

Choose several x-values around the expected center of the graph.
Calculate the corresponding y-values.
Plot the ordered pairs.
Use symmetry to check matching points on opposite sides of the axis.
Draw a smooth parabola through the points.

For $f (x) = x^{2} - 4$ , the points $(- 2, 0)$ , $(- 1, - 3)$ , $(0, - 4)$ , $(1, - 3)$ , and $(2, 0)$ reveal a vertex at $(0, - 4)$ and symmetry about $x = 0$ .

Method B: Use Vertex Form

For $f (x) = a (x - h)^{2} + k$ :

Plot the vertex $(h, k)$ .
Draw or imagine the axis of symmetry $x = h$ .
Use the sign of $a$ to decide whether the graph opens upward or downward.
Use the magnitude of $a$ to judge how quickly the graph rises or falls.
Plot one or more points on one side and reflect them across the axis.

Method C: Use Standard Form

For $f (x) = a x^{2} + b x + c$ :

Find the axis of symmetry with $x = - \frac{b}{2 a}$ .
Substitute that x-value into the function to find the vertex.
Plot the y-intercept $(0, c)$ .
Reflect the y-intercept across the axis of symmetry when useful.
Find x-intercepts if they are easy to calculate.
Sketch the smooth curve.

Method D: Use Roots and the Vertex

If the roots are known, plot the x-intercepts first. The axis of symmetry is halfway between two distinct real roots. Then find the vertex by substituting the axis x-value into the function.

Transformations of the Parent Parabola

Vertex form $f (x) = a (x - h)^{2} + k$ is especially useful for describing transformations of $y = x^{2}$ .

Horizontal shift: $h$ moves the graph right when $h > 0$ and left when $h < 0$ . Because the form uses $x - h$ , the sign inside the parentheses can look opposite to the movement.

Vertical shift: $k$ moves the graph up when $k > 0$ and down when $k < 0$ .

Reflection: If $a < 0$ , the graph is reflected across the x-axis.

Vertical stretch or compression: If $| a | > 1$ , the parabola appears narrower. If $0 < | a | < 1$ , it appears wider.

Compare these examples:

$y = (x - 2)^{2} + 1$ : shift right 2 and up 1.
$y = - (x + 3)^{2}$ : shift left 3 and reflect across the x-axis.
$y = 3 x^{2}$ : vertical stretch, so the graph is narrower.
$y = \frac{1}{4} x^{2}$ : vertical compression, so the graph is wider.

Roots, the Discriminant, and the Graph

For $a x^{2} + b x + c = 0$ , the quadratic formula is

$x = \frac{- b \pm \sqrt{b^{2} - 4 a c}}{2 a}$ .

The expression $b^{2} - 4 a c$ is the discriminant. Its sign predicts how many real x-intercepts the graph has.

If $b^{2} - 4 a c > 0$ , there are two distinct real roots and two x-intercepts.
If $b^{2} - 4 a c = 0$ , there is one repeated real root and the parabola touches the x-axis at the vertex.
If $b^{2} - 4 a c < 0$ , there are no real roots and no x-intercepts.

A graph therefore gives visual information about the solutions of a quadratic equation. Algebra and geometry are describing the same relationship in different ways.

Worked Example: From Equation to Graph

Consider $f (x) = - x^{2} + 6 x - 5$ .

First, identify $a = - 1$ , $b = 6$ , and $c = - 5$ . Because $a < 0$ , the graph opens downward.

The axis of symmetry is

$x = - \frac{6}{2 (- 1)} = 3$ .

Now calculate the vertex height:

$f (3) = - (3)^{2} + 6 (3) - 5 = 4$ .

So the vertex is $(3, 4)$ , and it is a maximum.

The y-intercept is $(0, - 5)$ .

To find x-intercepts, solve $- x^{2} + 6 x - 5 = 0$ . Factoring gives $- (x - 1) (x - 5) = 0$ , so the x-intercepts are $(1, 0)$ and $(5, 0)$ .

The roots are equally spaced from $x = 3$ , which confirms the axis of symmetry. The range is $y \leq 4$ .

Use the displayed example as a visual check: identify its opening direction first, then estimate its vertex and intercepts before calculating them from the equation shown on the graph.

Quadratic Models in Context

Quadratic functions can model situations in which a quantity rises and then falls, falls and then rises, or depends on a product of two changing quantities.

A common classroom example is vertical height over time. Ignoring air resistance, a projectile near Earth's surface can be modeled approximately by a quadratic function of time. The vertex represents the greatest height when the parabola opens downward. A positive root may represent the time when the object returns to a chosen reference height.

Another example comes from area. If the sides of a rectangle depend linearly on the same variable, multiplying the side lengths can produce a quadratic expression. The vertex can then represent a maximum or minimum area within the model.

When you use any mathematical model, check the domain that makes sense in context. A formula may accept every real x-value, but a real situation may allow only nonnegative time values or measurements within a physical range.

Common Errors and How to Catch Them

Confusing the sign of h: In $(x - h)^{2}$ , a positive $h$ shifts the graph right. For example, $(x - 4)^{2}$ has vertex x-coordinate 4.

Forgetting that a cannot be zero: If $a = 0$ , the x-squared term disappears, so the function is no longer quadratic.

Drawing a pointed vertex: A parabola is a smooth curve, not a V-shape.

Using roots as y-values: Roots are x-values where the function equals zero.

Ignoring symmetry: Points equally far from the axis of symmetry must have the same y-value.

Using an unrealistic domain in an application: Interpret only the x-values that make sense for the situation.

Interactive Tasks

Quiz: Test Your Knowledge

What is the graph of a quadratic function called? (A parabola) (!A circle) (!A line) (!A hyperbola)

What is the vertex of f of x equals 2 times x minus 3 squared minus 5? (3, -5) (!-3, -5) (!3, 5) (!-5, 3)

What happens when the leading coefficient is negative? (The parabola opens downward) (!The parabola becomes a line) (!The parabola always has two roots) (!The vertex moves to the origin)

Which expression gives the x-coordinate of the vertex in standard form? (-b divided by 2a) (!b divided by 2a) (!-c divided by 2a) (!a divided by 2b)

In standard form, which coefficient gives the y-intercept value? (c) (!a) (!b) (!a plus b)

What do real roots of a quadratic function represent on its graph? (x-intercepts) (!y-intercepts) (!axes of symmetry) (!vertical shifts)

What does a positive discriminant mean for a real quadratic graph? (It has two distinct x-intercepts) (!It has exactly one x-intercept) (!It has no x-intercepts) (!It must open downward)

How does y equals x minus 4 squared compare with y equals x squared? (It is shifted 4 units right) (!It is shifted 4 units left) (!It is shifted 4 units up) (!It is shifted 4 units down)

If an upward-opening parabola has vertex height k, what is its range? (y is greater than or equal to k) (!y is less than or equal to k) (!x is greater than or equal to k) (!x is less than or equal to k)

For f of x equals x squared minus 4x plus 3, what is the axis of symmetry? (x equals 2) (!x equals -2) (!x equals 3) (!x equals 4)

Memory Game

Vertex	The turning point that gives a minimum or maximum
Axis of symmetry	The vertical line that divides a parabola into mirror halves
Leading coefficient	The coefficient that controls opening direction and vertical stretch
Root	An x-value where the function equals zero
Y-intercept	The point where a graph crosses the vertical axis
Discriminant	The expression that predicts the number of real roots

Drag and Drop

Match the correct terms.	Topic
Vertex form	Equation that displays the turning point directly
Standard form	Equation that displays the constant term directly
Factored form	Equation that can display real zeros directly
Upward opening	Graph produced by a positive leading coefficient
Downward opening	Graph produced by a negative leading coefficient

Match each description with the correct representation or graph feature.

Crossword Puzzle

Parabola	What is the curved graph of a quadratic function called?
Vertex	What is the turning point of a parabola called?
Symmetry	What property makes the two sides of a parabola mirror each other?
Intercepts	What are points where a graph crosses or touches an axis called?
Discriminant	What expression predicts the number of real roots?
Transformation	What is a shift, reflection, stretch, or compression of a graph called?

LearningApps

Cloze Text

A quadratic function has highest variable exponent

. Its graph is called a

. The turning point of the graph is the

. A vertical line through the vertex is the axis of

. In standard form, the constant term gives the

. Real roots correspond to

on the graph. A positive leading coefficient makes the graph open

. A negative leading coefficient makes it open

. Vertex form reveals the coordinates of the

directly. The expression b squared minus 4ac is the

. A positive discriminant predicts

distinct real roots. A quadratic model should be interpreted only over a contextually meaningful

.

Open-Ended Tasks

Easy

Parabola Photo Hunt: Find or photograph four real objects or designs that look approximately parabolic, label where a vertex and symmetry axis might be, and explain why the shape is only an approximation of a mathematical graph.
Table-to-Graph Poster: Choose a simple quadratic rule, build a value table with at least seven points, draw the graph by hand, and annotate its vertex, axis of symmetry, opening direction, and intercepts.
Vertex Storyboard: Create a four-panel visual explanation showing how to read the vertex from vertex form and how the signs of h and k move the parent parabola.
Coefficient Slider Investigation: Use a graphing tool to vary the leading coefficient while keeping the vertex fixed, record at least six observations, and write a short rule describing changes in direction and width.

Standard

Projectile Model Investigation: Record or use safe sample data for the height of a tossed lightweight object, fit or select a reasonable quadratic model, and interpret its vertex and meaningful intercepts without extending the model beyond a sensible time interval.
Mathematics Interview: Interview a teacher, engineer, designer, programmer, or technician about where curved graphs or optimization appear in their work, then compare the interview examples with the features of a quadratic function.
Transformation Explainer Video: Produce a two- to four-minute video that starts with y equals x squared and demonstrates a shift, reflection, stretch, and compression using graphs and clear spoken explanations.
Parabolic Design Walk: Visit a school, park, bridge, sports area, or public building with permission, document possible parabolic shapes, and analyze one image by overlaying or sketching a coordinate system and an approximate quadratic graph.

Advanced

Quadratic Regression Investigation: Collect at least eight paired measurements from a suitable experiment or simulation, compare a quadratic model with a linear model, and justify which model better represents the visible pattern over the measured interval.
Optimization Design Challenge: Create a realistic area, revenue, or trajectory problem that leads to a quadratic function, determine the vertex, explain its meaning, and state any domain restrictions that make the answer realistic.
Discriminant Visual Proof Sketch: Build three carefully chosen quadratic examples with positive, zero, and negative discriminants, graph each one, and explain visually why the sign of the discriminant matches the number of real x-intercepts.
Multiple Representations Portfolio: Select one quadratic function and present it in standard form, vertex form, factored form when possible, a table, and a graph, then write a comparison explaining what information each representation reveals most efficiently.

Learning Assessment

Graph Reconstruction: You are given only a vertex, one additional point, and an opening direction; construct a possible quadratic function, graph it, and justify each parameter in your equation.
Error Analysis: Analyze a graph where a learner placed the vertex on the wrong side because of the sign in vertex form, identify the exact misconception, correct the graph, and create a rule that would prevent the error.
Model Interpretation: Given a quadratic height model for a safe classroom simulation, identify the meaningful domain, interpret the vertex and any positive root, and explain which algebraic results should not be interpreted physically.
Representation Choice: For three different tasks such as finding a y-intercept, locating a vertex, and reading roots, choose the most useful quadratic form for each task and defend your choices.
Parameter Comparison: Compare two quadratic functions that share the same vertex but have different leading coefficients, predict how their graphs differ before plotting, and then use a graph to evaluate your prediction.
Transfer Problem: Design a rectangular-area situation that produces a quadratic relationship, determine whether the vertex gives a maximum or minimum, and explain how the graph supports your conclusion.

Evidence of Learning

Strong evidence of learning includes both correct mathematical results and clear reasoning.

Knowledge: You can define a quadratic function, parabola, vertex, axis of symmetry, intercept, root, and discriminant in your own words.
Graphing skill: You can create an accurate quadratic graph from a table, standard form, vertex form, or factored form and check the result using symmetry.
Algebraic skill: You can find the vertex, intercepts, axis of symmetry, and discriminant and connect each calculation to a visible feature of the graph.
Representation skill: You can move between equations, tables, graphs, and verbal descriptions without losing the meaning of key features.
Product evidence: Your posters, investigations, models, videos, or portfolios use accurate graphs, readable labels, appropriate domains, and explanations that another learner can follow.
Transfer achievement: You can recognize when a new situation may be modeled quadratically, interpret the vertex or roots in context, and identify limits of the model.

OERs on the Topic

The English Wikipedia article on quadratic functions provides a useful reference for the standard, factored, and vertex forms, roots, and graph features.

Linked Learning Areas

The central idea is that a quadratic equation and its graph describe the same mathematical structure. The equation form can make some features easy to read, while the graph makes symmetry, extrema, intercepts, and overall behavior visible. Moving confidently among these representations is a key algebra skill.

Graphs of Quadratic Functions