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Algebra - Functions and graphs

Grade 7IGCSE

Review the key concepts, formulae, and examples before starting your quiz.

🔑Concepts

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The Cartesian Plane: Understanding the x-axis, y-axis, and plotting points in all four quadrants using coordinates (x,y)(x, y).

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Function Notation: Understanding that a function takes an input (xx) and produces an output (yy or f(x)f(x)).

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Linear Functions: Graphs that form a straight line, typically expressed in the form y=mx+cy = mx + c.

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Gradient (Slope): A measure of the steepness of a line, calculated as the 'rise over run'.

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Y-intercept: The point where the graph crosses the y-axis (where x=0x=0).

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Horizontal and Vertical Lines: Equations of the form x=kx = k (vertical) and y=ky = k (horizontal).

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Tables of Values: Creating a set of coordinates by substituting xx values into an equation to plot a graph.

📐Formulae

y=mx+cy = mx + c (Equation of a straight line)

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} (Gradient formula)

y−y1=m(x−x1)y - y_1 = m(x - x_1) (Point-gradient form)

M=(x1+x22,y1+y22)M = (\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}) (Midpoint of a line segment)

💡Examples

Problem 1:

Complete the table of values for the function y=3x−2y = 3x - 2 for xx values −1,0,1,2-1, 0, 1, 2.

Solution:

For x=−1:y=3(−1)−2=−5x = -1: y = 3(-1) - 2 = -5; For x=0:y=3(0)−2=−2x = 0: y = 3(0) - 2 = -2; For x=1:y=3(1)−2=1x = 1: y = 3(1) - 2 = 1; For x=2:y=3(2)−2=4x = 2: y = 3(2) - 2 = 4. The coordinates are (−1,−5),(0,−2),(1,1),(2,4)(-1, -5), (0, -2), (1, 1), (2, 4).

Explanation:

Substitute each given xx value into the equation y=3x−2y = 3x - 2 to find the corresponding yy value, then pair them as coordinates.

Problem 2:

Find the gradient of the line passing through the points A(2,5)A(2, 5) and B(4,13)B(4, 13).

Solution:

m=13−54−2=82=4m = \frac{13 - 5}{4 - 2} = \frac{8}{2} = 4.

Explanation:

Apply the gradient formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1} by identifying x1=2,y1=5x_1=2, y_1=5 and x2=4,y2=13x_2=4, y_2=13.

Problem 3:

Identify the gradient and y-intercept of the line with the equation 2y=6x+42y = 6x + 4.

Solution:

Divide by 2 to get y=3x+2y = 3x + 2. Gradient (mm) = 3, Y-intercept (cc) = 2.

Explanation:

To find mm and cc, the equation must first be rearranged into the standard form y=mx+cy = mx + c.

Problem 4:

What is the equation of the horizontal line that passes through the point (5,−3)(5, -3)?

Solution:

y=−3y = -3

Explanation:

A horizontal line has the same y-coordinate for every point on the line. Since it passes through (5,−3)(5, -3), the y-value is always −3-3.