🔮 Algebra Realm · Graphs

y-intercept

y-intercept focuses on how to identify and interpret the point where a graph crosses the y-axis. In this lesson, focus on straight-line graphs connect coordinates, rates and equations.

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Understand y-intercept

y-intercept focuses on how to identify and interpret the point where a graph crosses the y-axis. In this lesson, focus on straight-line graphs connect coordinates, rates and equations.

Straight-line graphs connect coordinates, rates and equations. Gradient measures vertical change per horizontal step; the y-intercept records the value when x is zero. For y-intercept, the final written answer should make that exact relationship visible rather than hiding it inside an unexplained result.

Start here

y-intercept: Straight-line graphs connect coordinates, rates and equations. Substitute one plotted coordinate into the equation or count a second rise-and-run triangle. Keep the y-intercept representation visible until the final line.

Picture the idea

Plot draggable points on a coordinate plane, draw the line and compare its rise, run and axis crossing. Use the model to explain one change you notice while working on y-intercept.

Check as you go

Substitute one plotted coordinate into the equation or count a second rise-and-run triangle. Write that check beside the final y-intercept answer.

Key vocabulary

coordinategradienty-interceptaxislinear relationshipyintercept

Rules and key facts

For y = 2x + 5, the y-intercept is 5.

  • Label and read both axes.
  • Plot or identify ordered pairs as x first, then y.
  • Use rise divided by run for gradient or read the y-axis crossing for intercept.
  • Check the relationship with a second point or a table value. Record the check explicitly for y-intercept.

Step-by-step method

  1. Label and read both axes.
  2. Plot or identify ordered pairs as x first, then y.
  3. Use rise divided by run for gradient or read the y-axis crossing for intercept.
  4. Check the relationship with a second point or a table value. Record the check explicitly for y-intercept.

What you need first

  • Recognise the vocabulary: coordinate, gradient, y-intercept.
  • Be able to explain the purpose of y-intercept before calculating.
  • Keep the relevant values, units and representation visible while you work.

Real-world use

  • Travel graphs
  • Fixed fees and rates

Visual / interactive

See the idea, then move it around

Skip to Practice

Plot draggable points on a coordinate plane, draw the line and compare its rise, run and axis crossing. Use the model to explain one change you notice while working on y-intercept.

Interactive maths model Connected to this topic; move controls, check outputs, then earn XP only from verified actions.
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Worked examples

Examples, methods and exam thinking

Level 1 · Foundation

Understand the idea with small numbers, one representation and one clear step.

Level 2 · Secure

Use the standard Year 8 method with mixed examples and normal wording.

Level 3 · Challenge

Handle multi-step or less familiar questions and explain choices.

Level 4 · Exam-style

Solve a worded question, show reasoning, check accuracy and write a final sentence.

Foundation example

Build confidence

Given information: y-intercept — State the y-intercept of y = 3x + 2. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 2. Final answer: 2. Check: substitute or compare with the original information to confirm the result fits the question.

  1. Label and read both axes.
  2. Plot or identify ordered pairs as x first, then y.
  3. Use rise divided by run for gradient or read the y-axis crossing for intercept.
Secure example

Use the normal method

Given information: y-intercept — State the y-intercept of y = 3x + 14. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 14. Final answer: 14. Check: substitute or compare with the original information to confirm the result fits the question.

Check: Interpret that starting or fixed value in context.

Challenge example

Stretch the idea

Given information: y-intercept — State the y-intercept of y = 3x + 13. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 13. Final answer: 13. Check: substitute or compare with the original information to confirm the result fits the question.

Try explaining why each step works before checking the answer.

Exam-style example

Show your reasoning

Given information: y-intercept — State the y-intercept of y = 3x + 12. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 12. Final answer: 12. Check: substitute or compare with the original information to confirm the result fits the question.

Exam tip: Write the intercept as (0, c) when coordinates are requested.

Common mistakes

  • Reading coordinates in the wrong order. This is a key trap when answering y-intercept questions.
  • Using horizontal change divided by vertical change for gradient.

How to check your answer

Substitute one plotted coordinate into the equation or count a second rise-and-run triangle. Write that check beside the final y-intercept answer.

Extension challenge

Create a y-intercept problem with a tempting incorrect answer. Solve it, apply the check, and explain exactly where the incorrect method breaks down.

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y-intercept challenge

Use plot patrol controls to solve three checked y-intercept rounds. Solve at least two of three marked rounds and use feedback to correct any error.

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Challenge y-intercept Arcane Boss

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y-intercept Arcane Boss

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Study cards

Core idea

y-intercept: Straight-line graphs connect coordinates, rates and equations. Substitute one plotted coordinate into the equation or count a second rise-and-run triangle. Keep the y-intercept representation visible until the final line.

Tap to mark reviewed
Key vocabulary

coordinate · gradient · y-intercept · axis · linear relationship · y · intercept

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Rules

Label and read both axes. Plot or identify ordered pairs as x first, then y. Use rise divided by run for gradient or read the y-axis crossing for intercept. Check the relationship with a second point or a table value. Record the check explicitly for y-intercept.

Tap to mark reviewed
Formula / fact

For y = 2x + 5, the y-intercept is 5.

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Foundation example

Given information: y-intercept — State the y-intercept of y = 3x + 2. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 2. Final answer: 2. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Secure example

Given information: y-intercept — State the y-intercept of y = 3x + 14. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 14. Final answer: 14. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Challenge example

Given information: y-intercept — State the y-intercept of y = 3x + 13. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 13. Final answer: 13. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Exam-style example

Given information: y-intercept — State the y-intercept of y = 3x + 12. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 12. Final answer: 12. Check: substitute or compare with the original information to confirm the result fits the question.

Tap to mark reviewed
Common mistake

Reading coordinates in the wrong order. This is a key trap when answering y-intercept questions.

Tap to mark reviewed
Exam tip

For y-intercept, show the key representation before the final calculation. Use this final check: Substitute one plotted coordinate into the equation or count a second rise-and-run triangle.

Tap to mark reviewed
Real-world use

Travel graphs, Fixed fees and rates

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Checklist

I can explain y-intercept, use the method, check for mistakes, and answer an exam-style question.

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Flashcards

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Help for y-intercept

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Simple explanation

y-intercept: Straight-line graphs connect coordinates, rates and equations. Substitute one plotted coordinate into the equation or count a second rise-and-run triangle. Keep the y-intercept representation visible until the final line.

Think of y-intercept as a careful model: make the important values visible, change one thing at a time, and use the check to prove the answer fits.

Step-by-step breakdown

  1. Label and read both axes.
  2. Plot or identify ordered pairs as x first, then y.
  3. Use rise divided by run for gradient or read the y-axis crossing for intercept.
  4. Check the relationship with a second point or a table value. Record the check explicitly for y-intercept.

Hint 1

Start by naming the given information and the exact result required for y-intercept.

Hint 2

Label and read both axes.

Full worked solution

Given information: y-intercept — State the y-intercept of y = 3x + 2. Method choice: use the y-intercept method and show each step with the stated values. Calculation or reasoning: In y = mx + c, c is the y-intercept. Here c = 2. Final answer: 2. Check: substitute or compare with the original information to confirm the result fits the question.

Method: Label and read both axes. → Plot or identify ordered pairs as x first, then y. → Use rise divided by run for gradient or read the y-axis crossing for intercept. → Check the relationship with a second point or a table value. Record the check explicitly for y-intercept.

Common mistake warning

Reading coordinates in the wrong order. This is a key trap when answering y-intercept questions.

Choose a support button above when you need a nudge.

Mastery milestones

Badges reward learning, not locked clicking

  • I can explain y-intercept in my own words.
  • I can use these words accurately: coordinate, gradient, y-intercept.
  • I can follow the 4-step method without guessing.
  • I can avoid this mistake: Reading coordinates in the wrong order.
  • I can apply this check: Substitute one plotted coordinate into the equation or count a second rise-and-run triangle.
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