Graphs - X-o-Y Plane
Section: Algebra | Syllabus: Cambridge Lower Secondary Checkpoint Mathematics (0862)
Linear Functions in Words and Symbols
A linear function can be written explicitly, y = mx + c (where m is the gradient and c is the y-intercept), or implicitly, ax + by = c. A real situation can be translated into either form.
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Question: A plumber charges a 40 call-out fee, plus 25 per hour worked. Write a linear function for the total cost, y, of a job lasting x hours.
- Step 1: Cost of the hours worked = 25x
- Step 2: Add the fixed fee: y = 25x + 40
- Answer: y = 25x + 40
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Question: Rearrange 3x + 2y = 12 into the form y = mx + c.
- Step 1: Subtract 3x from both sides: 2y = 12 − 3x
- Step 2: Divide both sides by 2: y = 6 − 1.5x
- Answer: y = −1.5x + 6
Plotting a Linear Function from a Table of Values
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Question: Construct a table of values for y = 2x + 3, using x = −1, 0, 1, 2, then plot the graph.
- Step 1: x = −1: y = 2(−1) + 3 = 1
- Step 2: x = 0: y = 3. x = 1: y = 5. x = 2: y = 7
- Step 3: Plot the points (−1, 1), (0, 3), (1, 5), (2, 7) and join them with a straight line
- Answer: a straight line through those 4 points
Plotting an Implicit Linear Function (ax + by = c)
For an equation like ax + by = c, the fastest way to plot it is to find where the line crosses each axis.
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Question: Plot 2x + y = 8.
- Step 1: Find the y-intercept (x = 0): 2(0) + y = 8, so y = 8 → point (0, 8)
- Step 2: Find the x-intercept (y = 0): 2x + 0 = 8, so x = 4 → point (4, 0)
- Step 3: Plot (0, 8) and (4, 0), then join with a straight line
- Answer: a straight line through (0, 8) and (4, 0)
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Question: Copy and complete this table of values for 5x + 4y = 20, giving the missing x-value when y = 0, the missing y-value when x = 0, and the missing y-value when x = 2.
- Step 1 (y = 0): 5x + 4(0) = 20, so 5x = 20, x = 4 → point (4, 0)
- Step 2 (x = 0): 5(0) + 4y = 20, so 4y = 20, y = 5 → point (0, 5)
- Step 3 (x = 2): 5(2) + 4y = 20, so 10 + 4y = 20, 4y = 10, y = 2.5 → point (2, 2.5)
- Answer: (4, 0), (0, 5), (2, 2.5) - three points confirm the straight line
Plotting a Quadratic Function (y = x² ± a)
A quadratic function produces a symmetric, U-shaped curve called a parabola. Plot several points either side of the turning point to see the curve clearly.
The graph of y = x² − 4: a symmetric curve with its lowest point at (0, −4)
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Question: Construct a table of values for y = x² − 4, using x = −3, −2, −1, 0, 1, 2, 3.
- Step 1: x = −3: y = 9 − 4 = 5. x = −2: y = 4 − 4 = 0. x = −1: y = 1 − 4 = −3
- Step 2: x = 0: y = −4. By symmetry, x = 1, 2, 3 give y = −3, 0, 5
- Answer: the curve has its lowest point at (0, −4) and is symmetric about the y-axis
Remember: the square of a negative number is positive, e.g. (−3)² = 9, not −9 - this is why the curve is symmetric rather than lopsided.
The same rule works for any x-value, even ones outside the table you plotted - and it can be used backwards too, to find x from a given y.
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Question: Using the rule y = x² − 4, find the value of y when x = 6 (a value outside the original table).
- Step 1: Substitute x = 6 into the rule: y = 6² − 4
- Answer: y = 32
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Question: Using the rule y = x² − 4, find the value(s) of x when y = 21.
- Step 1: Substitute y = 21: 21 = x² − 4
- Step 2: Add 4 to both sides: x² = 25
- Step 3: Take the square root of both sides - remember a positive and a negative value both work: x = 5 or x = −5
- Answer: x = 5 or x = −5
Finding the Equation of a Straight Line from a Graph
To find a line's equation in the form y = mx + c: read the y-intercept, c, directly from where the line crosses the y-axis, and calculate the gradient using m = (change in y) ÷ (change in x) between any two points on the line. The sign of the gradient matches the direction of the line: moving left to right, a line that rises has a positive gradient, and a line that falls has a negative gradient.
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Question: A line passes through (0, 4) and (2, 8). Find its equation.
- Step 1: Gradient = (8 − 4)/(2 − 0) = 4/2 = 2
- Step 2: The line crosses the y-axis at (0, 4), so c = 4
- Answer: y = 2x + 4
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Question: A line passes through (0, 2) and (4, 4). Find its equation.
- Step 1: Gradient = (4 − 2)/(4 − 0) = 2/4 = 1/2
- Step 2: The line crosses the y-axis at (0, 2), so c = 2
- Answer: y = (1/2)x + 2 (a fractional gradient)
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Question: A line has gradient −3 and y-intercept 5. Write its equation.
- Step 1: m = −3, c = 5
- Answer: y = −3x + 5 (a negative gradient)
Parallel Lines and Direct Proportion
Parallel lines never meet, because they rise or fall at exactly the same rate - they have the same gradient but different y-intercepts. A line that passes through the origin, (0, 0), represents direct proportion: the gradient is the constant rate connecting the two quantities.
Three parallel lines: same gradient, different y-intercepts
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Question: Write down the equations of two different lines that are parallel to y = 4x − 1.
- Step 1: Parallel lines share the same gradient, so both new lines must also have gradient 4
- Step 2: Choose any y-intercept other than −1
- Answer: e.g. y = 4x + 2 and y = 4x − 9
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Question: Kwame converts distances from kilometres to miles. His graph is a straight line through the origin, passing through (80, 50) - meaning 80 km is equivalent to 50 miles. Find the equation linking miles, y, and kilometres, x, then use it to convert 200 km to miles.
- Step 1: Since the line passes through the origin, this is direct proportion: gradient = 50 ÷ 80 = 0.625
- Step 2: Equation: y = 0.625x
- Step 3: Substitute x = 200: y = 0.625 × 200 = 125
- Answer: y = 0.625x; 200 km is equivalent to 125 miles
Reading and Interpreting Graphs
When reading a graph, pay attention to what each axis represents, and what different features of the graph mean in context.
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