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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

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

For Teachers

Learning Objectives
  • Identify the coefficients aa, bb, and cc in a quadratic function
  • Determine whether a parabola opens upward or downward based on the sign of aa
  • 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

Remediation: Show multiple parabolas with different vertices. Have students calculate vertices for y=x2y = x^2, y=x2+3y = x^2 + 3, and y=(x−2)2y = (x-2)^2 to see how they differ.

The coefficient cc is the vertex's y-coordinate

Remediation: Compare y=x2+4y = x^2 + 4 (vertex at (0,4)(0, 4)) with y=x2−2x+4y = x^2 - 2x + 4 (vertex at (1,3)(1, 3)). The y-intercept and vertex are only the same when b=0b = 0.

Larger ∣a∣|a| makes the parabola wider

Remediation: Graph y=x2y = x^2, y=2x2y = 2x^2, and y=0.5x2y = 0.5x^2 on the same axes. Larger ∣a∣|a| actually makes it narrower (steeper).

Differentiation Ideas

For Struggling Students:

  • • Start with parabolas in the form y=x2+cy = x^2 + c (no bb 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 y=a(x−h)2+ky = a(x-h)^2 + k 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:
f(x)=ax2+bx+cf(x) = ax^2 + bx + c
where aa, bb, and cc are constants and a≠0a \neq 0.
The graph of a quadratic function is called a parabola - a symmetric U-shaped curve.
Key Features of a Parabola:
FeatureDescriptionFormula
VertexThe highest or lowest point(−b2a,f(−b2a))\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)
Axis of SymmetryVertical line through the vertexx=−b2ax = -\frac{b}{2a}
DirectionOpens up if a>0a > 0, down if a<0a < 0Sign of aa
Y-interceptWhere the graph crosses the y-axis(0,c)(0, c)
X-intercepts (Roots)Where the graph crosses the x-axisSolve ax2+bx+c=0ax^2 + bx + c = 0

Worked Examples

Graph f(x)=x2−4x+3f(x) = x^2 - 4x + 3 and identify all key features.
1

Identify the coefficients

a=1a = 1, b=−4b = -4, c=3c = 3 → Parabola opens upward (since a>0a > 0)
2

Find the axis of symmetry

x=−b2a=−−42(1)=42=2x = -\frac{b}{2a} = -\frac{-4}{2(1)} = \frac{4}{2} = 2 → Axis of symmetry: x=2x = 2
3

Find the vertex

f(2)=(2)2−4(2)+3=4−8+3=−1f(2) = (2)^2 - 4(2) + 3 = 4 - 8 + 3 = -1 → Vertex: (2,−1)(2, -1)
4

Find the y-intercept

f(0)=(0)2−4(0)+3=3f(0) = (0)^2 - 4(0) + 3 = 3 → Y-intercept: (0,3)(0, 3)
5

Find the x-intercepts (roots)

x2−4x+3=0⇒(x−1)(x−3)=0x^2 - 4x + 3 = 0 \Rightarrow (x-1)(x-3) = 0 → X-intercepts: (1,0)(1, 0) and (3,0)(3, 0)
6

Plot points and draw the parabola

Plot vertex (2,−1)(2, -1), y-intercept (0,3)(0, 3), x-intercepts (1,0)(1, 0) and (3,0)(3, 0) → U-shaped curve opening upward

Common Mistakes

Forgetting the negative sign when calculating the axis of symmetry

Why it's wrong: The formula x=−b2ax = -\frac{b}{2a} has a negative sign that's easy to miss, especially when bb is already negative.

Correct: Always write the formula with the negative sign first: x=−b2ax = -\frac{b}{2a}. If b=−4b = -4 and a=1a = 1, then x=−−42=42=2x = -\frac{-4}{2} = \frac{4}{2} = 2.

Confusing the vertex with the y-intercept

Why it's wrong: The y-intercept (0,c)(0, c) is often easier to find, but it's not the vertex unless the axis of symmetry is x=0x = 0.

Correct: The vertex is at (−b2a,f(−b2a))\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right). 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 aa determines the direction.

Correct: Always check the sign of aa first: a>0a > 0 opens up (minimum), a<0a < 0 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 Δ=b2−4ac\Delta = b^2 - 4ac: if Δ>0\Delta > 0, two x-intercepts; if Δ=0\Delta = 0, one; if Δ<0\Delta < 0, 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 h(t)=−5t2+20t+1.5h(t) = -5t^2 + 20t + 1.5 meters after tt 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 h(t)=−4.9t2+14.7th(t) = -4.9t^2 + 14.7t meters.

What is the maximum height reached by the ball?

Step 1: Write the mathematical expression

Find when the ball reaches maximum height using t=−b2at = -\frac{b}{2a}:

Business Revenue

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

Example:

If a company's revenue is R(p)=−2p2+100pR(p) = -2p^2 + 100p where pp 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 R(p)=−50p2+400pR(p) = -50p^2 + 400p euros, where pp 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 f(x)=ax2+bx+cf(x) = ax^2 + bx + c where a≠0a \neq 0
  • 2The graph is a parabola: U-shaped if a>0a > 0, inverted U if a<0a < 0
  • 3The axis of symmetry is the vertical line x=−b2ax = -\frac{b}{2a}
  • 4The vertex is at (−b2a,f(−b2a))\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right) - either a minimum or maximum
  • 5The y-intercept is (0,c)(0, c)
  • 6The discriminant Δ=b2−4ac\Delta = b^2 - 4ac 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 f(x)=ax2+bx+cf(x) = ax^2 + bx + c and produces outputs for any input. A quadratic equation is ax2+bx+c=0ax^2 + bx + c = 0 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 Δ=b2−4ac\Delta = b^2 - 4ac is negative, the parabola never crosses the x-axis. For example, f(x)=x2+1f(x) = x^2 + 1 has its lowest point at (0,1)(0, 1), which is above the x-axis.

Glossary

Quadratic function
A polynomial function of degree 2 in the form f(x)=ax2+bx+cf(x) = ax^2 + bx + c where a≠0a \neq 0
Parabola
The U-shaped curve that is the graph of a quadratic function
Vertex
The highest or lowest point on a parabola, located at (−b2a,f(−b2a))\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)
Axis of symmetry
The vertical line x=−b2ax = -\frac{b}{2a} that divides the parabola into two mirror-image halves
Discriminant
The value Δ=b2−4ac\Delta = b^2 - 4ac that determines the number of x-intercepts
Roots (zeros)
The x-values where f(x)=0f(x) = 0, also called x-intercepts

Formula Card

Standard Form

f(x)=ax2+bx+cf(x) = ax^2 + bx + c

The general form of a quadratic function

Axis of Symmetry

x=−b2ax = -\frac{b}{2a}

Vertical line through the vertex

Vertex

(−b2a,f(−b2a))\left(-\frac{b}{2a}, f\left(-\frac{b}{2a}\right)\right)

The highest or lowest point

Discriminant

Δ=b2−4ac\Delta = b^2 - 4ac

Determines number of roots

Quadratic Formula

x=−b±b2−4ac2ax = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}

Finds x-intercepts