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.

Advanced25 min16 problems
Practice Problems

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

Try it now

For the function f(x)=2x2+3x−5f(x) = 2x^2 + 3x - 5, what is the value of coefficient aa?

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.

Interactive Visual

Function graph

-4-202468-10-50510xy(1, 0)(3, 0)(2, -1)

Tap or hover over the graph to read off values.

  • Axis of symmetry x = 2
  • Δ = b² − 4ac = 4
f(x) = ax² + bx + c
  • Zero (x-intercept): (1, 0)
  • Zero (x-intercept): (3, 0)
  • Vertex: (2, -1)

Interactive Sandbox

Interactive Grapher

-10-50510-10-50510xy

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x^2 - 4x + 3

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Practice Problems

16 problems
Problem 1 of 16
Easy
For the function f(x)=2x2+3x−5f(x) = 2x^2 + 3x - 5, what is the value of coefficient aa?

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.

At what time tt (in seconds) does the ball reach its maximum height?

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 dollars, 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 dollars, 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

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).
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).
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.
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

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