Linear Functions from Two Points: Slope, Equation and UPCAT Practice

TEACHER ABI UPCAT MATHEMATICS

Linear Functions from Two Points

Find the constant rate of change, build the rule, and interpret what slope and intercept mean in real situations.

5-10 minute lesson27 original questionsAdaptive practiceSaves progress

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Linear Functions

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Two points determine one nonvertical line

For points (x₁,y₁) and (x₂,y₂), the slope is the change in y divided by the change in x. Once you know the slope, use either point to find the intercept or write point-slope form.

Slope

m=y₂−y₁x₂−x₁. Keep the subtraction order consistent.

Equation

Use y−y₁=m(x−x₁), or substitute into y=mx+b to find b.

DO IT FAST

Find the rate, then move from a known point

For a requested x-value, you often do not need the full equation. Compute the slope, count how far x moved from a known point, and add slope × change in x to the known y-value.

Why it works

A linear function changes by the same amount in y for every one-unit change in x. The shortcut is simply the slope definition used forward.

WORKED EXAMPLES

Five forms you should recognize

1. Find slope

Through (2,5) and (6,13): m=13−56−2=2.

2. Find the equation

Using (2,5): 5=2(2)+b, so b=1 and y=2x+1.

3. Predict directly

If cost is ₱50 at 20 pages and ₱80 at 35 pages, the rate is ₱2 per page; 25 pages cost ₱60.

4. Interpret slope

In C=12n+40, 12 is the added cost per unit.

5. Compare models

The larger slope grows faster; the intercept controls the starting value.

COMMON TRAPS

Check before you commit

• Mixed subtraction order: reverse both numerator and denominator or neither.
• Using yx: slope is change in y over change in x, not a single coordinate ratio.
• Forgetting the intercept: a proportional rule is not assumed unless the line passes through the origin.
• Confusing variables: identify which quantity depends on the other.
• Extrapolation: predictions far outside given data rely on the linear assumption.
• Vertical line: equal x-coordinates give undefined slope and no function y=f(x).

FIVE-FORM SKILL CHECK

Do you need the lesson-or just practice?

One original question in each form recommends your next step. It does not yet verify mastery.

CHOOSE YOUR PRACTICE

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Foundations

Build the core procedure with immediate explanations.

Core Practice

Use mixed forms with less scaffolding.

UPCAT-Style Transfer

Apply the competency in unfamiliar representations.

FRESH MASTERY CHECK

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QUICK ANSWERS

Linear Functions from Two Points FAQ

Can I predict without finding b?

Yes. Move from a known point using change in y = slope × change in x.

What does a negative slope mean?

The output decreases as the input increases.

When is y=mx enough?

Only when the line passes through the origin, so b=0.

What if both points have the same x-coordinate?

The line is vertical, its slope is undefined, and it is not a function of x.

RELATED COMPETENCIES

Continue your mathematics review.

SlopeEquation of a LineDirect VariationSystems of EquationsGraph Interpretation

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