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Algebra - Introduction to Linear Graphs and the Cartesian Plane

Grade 8IB

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

🔑Concepts

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The Cartesian Plane consists of two perpendicular number lines: the horizontal xx-axis and the vertical yy-axis. Their intersection point (0,0)(0,0) is called the origin. Each point is identified by an ordered pair (x,y)(x, y), where xx indicates horizontal distance and yy indicates vertical distance.

A Cartesian plane showing the x and y axes with two labeled points P(3, 2) and Q(-4, -3).
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Linear graphs are straight lines represented by the equation y=mx+cy = mx + c. The coefficient mm represents the gradient (steepness), calculated as riserun\frac{\text{rise}}{\text{run}}. The constant cc is the yy-intercept, where the line crosses the yy-axis.

A graph of the line y = x + 1 showing the y-intercept at (0,1).
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The gradient (slope) of a line determines its orientation. A positive gradient (m>0m > 0) slants upwards from left to right, while a negative gradient (m<0m < 0) slants downwards. A horizontal line has m=0m = 0, and a vertical line has an undefined gradient.

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To plot a linear graph from an equation, create a table of values for xx and solve for yy, then plot the resulting coordinate pairs and connect them with a straight line.

📐Formulae

y=mx+cy = mx + c

m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}

m=riserunm = \frac{\text{rise}}{\text{run}}

ax+by=dax + by = d

💡Examples

Problem 1:

Find the gradient (mm) and the yy-intercept (cc) for the line given by the equation 3x+y=73x + y = 7, and state the coordinates of the yy-intercept.

Solution:

  1. Rearrange the equation into the form y=mx+cy = mx + c by subtracting 3x3x from both sides: y=−3x+7y = -3x + 7
  2. Identify the coefficient of xx as the gradient: m=−3m = -3
  3. Identify the constant term as the yy-intercept: c=7c = 7
  4. The coordinates of the yy-intercept are (0,7)(0, 7).

Explanation:

To identify key features of a linear graph, it is easiest to convert the equation into gradient-intercept form. The gradient tells us the line drops 3 units for every 1 unit it moves right.

Problem 2:

Calculate the gradient of the line passing through the points A(2,5)A(2, 5) and B(6,13)B(6, 13).

Solution:

  1. Identify the coordinates: (x1,y1)=(2,5)(x_1, y_1) = (2, 5) and (x2,y2)=(6,13)(x_2, y_2) = (6, 13).
  2. Use the gradient formula: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}
  3. Substitute the values: m=13−56−2m = \frac{13 - 5}{6 - 2}
  4. Simplify the numerator and denominator: m=84m = \frac{8}{4}
  5. Calculate the final result: m=2m = 2.

Explanation:

The gradient is the ratio of the change in vertical height (rise) to the change in horizontal distance (run). A gradient of 2 means the line is rising.

Problem 3:

Identify the coordinates of the points MM and NN shown on the grid, and determine the gradient of the line segment MNMN.

A line segment connecting point M(-2, 1) and N(2, 4) on a coordinate grid.

Solution:

  1. From the grid, point MM is at (−2,1)(-2, 1) and point NN is at (2,4)(2, 4).
  2. Use the gradient formula: m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}.
  3. m=4−12−(−2)=32+2=34m = \frac{4 - 1}{2 - (-2)} = \frac{3}{2 + 2} = \frac{3}{4}.
  4. The gradient is 0.750.75.

Explanation:

The gradient represents how much the yy-value increases for every unit increase in xx. Here, for every 44 units right, the line goes 33 units up.

Problem 4:

Identify the xx-intercept and yy-intercept from the graph of the linear equation shown. Then, use these intercepts to calculate the gradient (mm) of the line.

A linear graph passing through the y-axis at 2 and the x-axis at 4.

Solution:

  1. Identify the Intercepts: From the graph, the line crosses the xx-axis at (4,0)(4, 0). Therefore, the xx-intercept is 44. The line crosses the yy-axis at (0,2)(0, 2). Therefore, the yy-intercept is 22.

  2. Identify Coordinates: Let (x1,y1)=(0,2)(x_1, y_1) = (0, 2) and (x2,y2)=(4,0)(x_2, y_2) = (4, 0).

  3. Calculate Gradient: Using the formula m=y2−y1x2−x1m = \frac{y_2 - y_1}{x_2 - x_1}: m=0−24−0m = \frac{0 - 2}{4 - 0} m=−24m = \frac{-2}{4} m=−12m = -\frac{1}{2}

The gradient of the line is −12-\frac{1}{2}.

Explanation:

The xx-intercept is the point where the graph intersects the xx-axis (y=0y=0), and the yy-intercept is where it intersects the yy-axis (x=0x=0). The gradient represents the rate of change, calculated as the change in yy over the change in xx. Since the line slopes downwards from left to right, the gradient is negative.