214. Connecting Graphs and Solutions of Quadratics

Learning Intentions

• Interpret roots as intercepts on a Draw.
• Compare algebraic solutions with graphical solutions.
• Explain how factorised form reveals the roots.

Pre-requisite Summary

• Solve quadratic equations by factorisation.
• Apply the null factor law to find solutions.
• Recognise quadratic graphs as parabolas.
• Identify x-intercepts from graphs.
• Plot points on the Cartesian plane.
• Substitute values into algebraic expressions.
• Understand factorised form of quadratics.

Worked Examples

Worked Example 1

Consider the quadratic function:

a) Factorise the expression.

b) Find the roots.

c) Explain what the roots represent on the graph.

Worked Example 2

Solve and interpret graphically:

a) Solve algebraically.

b) Describe the x-intercepts of the graph of .

Worked Example 3

Given:

a) Find the roots.

b) Explain how the factorised form shows the x-intercepts.

c) Sketch the key features of the graph.

Worked Example 4

Compare methods:

Solve:

a) Solve algebraically.

b) Sketch or describe the graph.

c) Check that the roots match the x-intercepts.

Worked Example 5

A quadratic function is:

a) Factorise.

b) Find the roots.

c) Explain how these relate to the graph crossing the x-axis.

Problems

Problem 1

Factorise and interpret:

a) Factorise

b) Find roots

c) State the x-intercepts

Problem 2

Solve and interpret:

a) Solve algebraically

b) Describe intercepts on the graph of

Problem 3

Given:

a) Find the roots

b) Explain what they mean on a graph

c) Sketch key features

Problem 4

Compare algebraic and graphical results:

a) Solve algebraically

b) Describe the graph’s x-intercepts

Problem 5

Given:

a) Factorise

b) Find roots

c) Explain how the graph behaves at the x-axis

Exercises

Understanding and Fluency

Exercise 1

Find the roots of:

Exercise 2

Factorise and find roots:

Exercise 3

Find the x-intercepts of:

Exercise 4

Solve and interpret:

Exercise 5

Given:

find the roots.

Exercise 6

Match the roots to the intercepts for:

Exercise 7

State the roots of:

Exercise 8

Explain what happens when a quadratic has no real roots in terms of a graph.

Exercise 9

Check algebraic solutions for:

Exercise 10

Explain the relationship between solutions and x-intercepts.

Reasoning

Exercise 11

Explain why factorised form makes it easy to find roots.

Exercise 12

A student says:

“The roots of a quadratic are the same as the y-intercepts.”

Explain the error.

Exercise 13

Describe how you can tell the number of x-intercepts from the equation.

Exercise 14

Explain why every root corresponds to a point where .

Problem-solving

Exercise 15

A projectile is modelled by:

a) Find roots

b) Interpret physically what the roots represent

Exercise 16

A bridge arch follows:

a) Find intercepts

b) Describe where it crosses the ground level

Exercise 17

A quadratic model is:

a) Find roots

b) Explain what is unusual about the graph’s x-intercepts

c) Interpret the meaning of this in context

Potential Misunderstandings

• Confusing x-intercepts with y-intercepts.
• Thinking roots are points instead of x-values where .
• Forgetting that graphs cross the x-axis only when .
• Assuming factorised form always shows intercepts without setting .
• Missing that repeated roots mean the graph touches but does not cross the x-axis.
• Misreading factorised expressions and changing signs incorrectly.
• Not linking algebraic solutions back to graphical meaning.
• Assuming all quadratics have two distinct intercepts.

Next: 215. Linear Parameters