Teacher Guide: Graphing Quadratic Functions (Parabolas)

Learn to graph quadratic functions, identify key features like vertex and axis of symmetry, and understand how coefficients affect the shape of parabolas.

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All practice problems on paper, with a separate answer key.

Worksheet Answer key

Class quiz

10 questions on Nonlinear Graphs. Students join with a name, you see everyone's score.

For Teachers

Learning Objectives

Identify the coefficients , , and in a quadratic function
Determine whether a parabola opens upward or downward based on the sign of
Calculate the axis of symmetry and vertex of a parabola
Find the y-intercept and x-intercepts of a quadratic function
Use the discriminant to determine the number of x-intercepts
Graph quadratic functions by plotting key features

Prerequisites

• Understanding of coordinate planes and plotting points
• Solving quadratic equations by factoring
• Evaluating expressions with exponents
• Understanding of symmetry

Discussion Starters

1. Why do you think satellite dishes are shaped like parabolas?
2. If you throw a ball, what factors would change the shape of its parabolic path?
3. How can you tell just by looking at an equation whether its graph will open up or down?
4. What real-world situation would have a parabola with no x-intercepts?

Common Misconceptions

The vertex is always at the origin

The coefficient is the vertex's y-coordinate

Larger makes the parabola wider

Differentiation Ideas

For Struggling Students:

• Start with parabolas in the form (no term) so the vertex is on the y-axis
• Use graphing technology to visualize before calculating
• Provide a step-by-step checklist for finding key features

For On-Level Students:

• Practice finding all key features from standard form
• Connect to solving quadratic equations
• Apply to word problems involving projectile motion

For Advanced Students:

• Explore vertex form and convert between forms
• Analyze how the discriminant relates to the graph
• Investigate families of parabolas that share certain features

Standards Alignment

HSF-IF.C.7a (CCSS.MATH.CONTENT.HSF.IF.C.7.A)
Graph quadratic functions and show intercepts, maxima, and minima
HSF-IF.B.4 (CCSS.MATH.CONTENT.HSF.IF.B.4)
Interpret key features of graphs and tables in terms of quantities
HSA-REI.B.4 (CCSS.MATH.CONTENT.HSA.REI.B.4)
Solve quadratic equations by inspection, factoring, completing the square, and the quadratic formula

Lesson Resources

visualInteractive Quadratic Explorer
Adjust sliders to see how a, b, and c affect the parabola
activityParabola Matching Game
Match equations to their graphs
worksheetKey Features Practice
Find vertex, axis of symmetry, and intercepts for various quadratics

Lesson Content

Everything students see: definition, examples, common mistakes, applications. Tap to open.

Definition

A quadratic function is a polynomial function of degree 2, written in the form:

where , , and are constants and .

The graph of a quadratic function is called a parabola - a symmetric U-shaped curve.

Key Features of a Parabola:

Feature	Description	Formula
Vertex	The highest or lowest point
Axis of Symmetry	Vertical line through the vertex
Direction	Opens up if , down if	Sign of
Y-intercept	Where the graph crosses the y-axis
X-intercepts (Roots)	Where the graph crosses the x-axis	Solve

Worked Examples

Graph and identify all key features.

Identify the coefficients

, , → Parabola opens upward (since )

Find the axis of symmetry

→ Axis of symmetry:

Find the vertex

→ Vertex:

Find the y-intercept

→ Y-intercept:

Find the x-intercepts (roots)

→ X-intercepts: and

Plot points and draw the parabola

Plot vertex , y-intercept , x-intercepts and → U-shaped curve opening upward

Answer: The parabola has vertex , axis of symmetry , opens upward, crosses the x-axis at and , and crosses the y-axis at .

Common Mistakes

Forgetting the negative sign when calculating the axis of symmetry

Why it's wrong: The formula has a negative sign that's easy to miss, especially when is already negative.

Correct: Always write the formula with the negative sign first: . If and , then .

Confusing the vertex with the y-intercept

Why it's wrong: The y-intercept is often easier to find, but it's not the vertex unless the axis of symmetry is .

Correct: The vertex is at . Calculate the x-coordinate first, then substitute to find the y-coordinate.

Thinking all parabolas open upward

Why it's wrong: Many students forget that the sign of determines the direction.

Correct: Always check the sign of first: opens up (minimum), opens down (maximum).

Assuming every parabola crosses the x-axis

Why it's wrong: Not all quadratic functions have real roots. The discriminant determines this.

Correct: Check : if , two x-intercepts; if , one; if , none.

Why It Matters

Quadratic functions appear everywhere in the real world:

Physics: The path of a thrown ball follows a parabola due to gravity
Engineering: Satellite dishes and car headlights use parabolic shapes to focus signals and light
Business: Revenue and profit models often involve quadratic functions
Architecture: Arches and bridges use parabolic curves for structural strength
Sports: The trajectory of a basketball shot, golf ball, or soccer kick is parabolic

Understanding parabolas helps you analyze projectile motion, optimize designs, and solve real-world problems!

Real World Applications

Projectile Motion

When you throw a ball, its height over time follows a parabolic path due to gravity.

Example:

A ball thrown upward has height meters after seconds. The vertex gives the maximum height, and the positive root gives when it lands.

1Try It Yourself

A soccer ball is kicked with height modeled by meters.

What is the maximum height reached by the ball?

Step 1: Write the mathematical expression

Find when the ball reaches maximum height using :

Business Revenue

Companies use quadratic functions to model how price changes affect revenue.

Example:

If a company's revenue is where is the price in euros, the vertex shows the price that maximizes revenue.

2Try It Yourself

A coffee shop's daily revenue is modeled by euros, where is the price per cup.

What price maximizes daily revenue?

Step 1: Write the mathematical expression

Use the vertex formula to find the optimal price:

Key Takeaways

1A quadratic function has the form where
2The graph is a parabola: U-shaped if , inverted U if
3The axis of symmetry is the vertical line
4The vertex is at - either a minimum or maximum
5The y-intercept is
6The discriminant determines the number of x-intercepts

Frequently Asked Questions

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

A quadratic function is and produces outputs for any input. A quadratic equation is and we solve it to find specific x-values (the roots).

Why is it called a parabola?

The word comes from the Greek 'parabole' meaning 'comparison' or 'placing side by side.' It was named by the Greek mathematician Apollonius around 200 BCE when studying conic sections.

Can a parabola have no x-intercepts?

Yes! When the discriminant is negative, the parabola never crosses the x-axis. For example, has its lowest point at , which is above the x-axis.

Glossary

Quadratic function: A polynomial function of degree 2 in the form where
Parabola: The U-shaped curve that is the graph of a quadratic function
Vertex: The highest or lowest point on a parabola, located at
Axis of symmetry: The vertical line that divides the parabola into two mirror-image halves
Discriminant: The value that determines the number of x-intercepts
Roots (zeros): The x-values where , also called x-intercepts

Formula Card

Standard Form

The general form of a quadratic function

Axis of Symmetry

Vertical line through the vertex

Vertex

The highest or lowest point

Discriminant

Determines number of roots

Quadratic Formula

Finds x-intercepts