How to plot a linear equation graph

Key points

Y equals m x plus c. An arrow points to y labelled y coordinate. An arrow points to m labelled gradient – this is highlighted pink. An arrow points to x labelled x coordinate. An arrow points to c labelled y intercept – this is highlighted green. — Image caption,
The graph of a oblique straight line is described using the equation, 𝒚 = 𝒎𝒙 + 𝒄

Graphs of two or more straight lines can be used to solve simultaneous linear equations.
The graph of a straight line can be described using an .
- lines are written as \(y = c\)
- lines are written as \(x = c\)
- are written as \(y = mx + c\)
\(m\) is a number which is a measure of the steepness of the line. This is the .
\(c\) is the number where the line crosses the \(y\)-axis. This is the \(y\).
The of the points on an oblique line are calculated by given values of \(x\) into the equation \(y = mx + c\)

Recognise and draw the lines 𝒚 = 𝒙 and 𝒚 = -𝒙

All the points on the line \(y = x\) have coordinates with equal values for \(x\) and \(y\)

To draw the line \(y = x\):
1. Plot points with coordinates where \(x\) and \(y\) are equal. Three points are sufficient, but more can be plotted.
2. Draw a line through the plotted points.

All the points on the line \(y = -x\) have coordinates with values for \(x\) and \(y\) that are equal in but with opposite signs.

If \(x\) is positive, \(y\) is negative. If \(x\) is negative, \(y\) is positive.

To draw the line \(y = -x\):
1. Plot points with coordinates where \(x\) and \(y\) have equal magnitude but opposite signs.
2. Draw a line through the plotted points.

Examples

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The straight line 𝒚 = 𝒙 passes through the origin. All the points on the line 𝒚 = 𝒙 have coordinates with equal values for 𝒙 and 𝒚
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To draw the line 𝒚 = 𝒙, plot points with coordinates where 𝒙 and 𝒚 are equal. Although three points are sufficient, more points may be plotted. Using well-spaced points, such as the coordinates (-7, -7), (0, 0) and (8, 8), the line may be accurately drawn.
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Next, draw a line through the plotted points. Label the line 𝒚 = 𝒙. The scales on the axes are equal, 𝒚 = 𝒙 is a diagonal line passing through the origin.
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The straight line, 𝒚 = -𝒙 passes through the origin. All the points on the line 𝒚 = -𝒙 have coordinates with values for 𝒙 and 𝒚 that are equal in magnitude but with opposite signs. If 𝒙 is positive, 𝒚 is negative. If 𝒙 is negative, 𝒚 is positive.
Image caption,
To draw the line 𝒚 = -𝒙, plot points with coordinates where 𝒙 and 𝒚 have equal magnitude but opposite signs. For example, the coordinates (-7, 7), (2, -2) and (9, -9) have 𝒙 and 𝒚 values that are equal in size, but with opposite signs, one being positive and the other negative.
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Draw a line through the plotted points. Label the line 𝒚 = -𝒙. The scales on the axes are equal, 𝒚 = -𝒙 is a diagonal line passing through the origin.

Question

One graph shows \(y = x\) and one shows \(y = -x\). Which graph shows \(y = x\)?

Two graphs, one labelled A and a second labelled B. Graph A showing the x axis and y axis increasing in units of five from minus ten to ten, intersecting at zero comma zero. A diagonal line passes through the origin, sloping down from left to right – this is highlighted orange. Graph B showing the x axis and y axis increasing in units of five from minus ten to ten, intersecting at zero comma zero. A diagonal line passes through the origin, sloping up from left to right – this is highlighted orange.

Draw the graph 𝒚 = 𝒎𝒙 + 𝒄 by creating a table of values

\(m\) is a number which measures the steepness of the line. This is known as the gradient.

\(c\) is the number where the line crosses the \(y\)-axis. This is the \(y\)-intercept.

To draw a graph of \(y = mx + c\) for given values of \(x\):
1. Use the given values for \(x\) to draw a table of values for \(x\) and \(y\)
2. each value of \(x\) into the equation to find the valueof \(y\). Each pair of values give a coordinate.
3. Use the coordinates to decide on that will take all the values of \(x\) and \(y\)
4. Plot the coordinates and draw a line through the points. Label the line with the equation.

Example

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Draw the graph of 𝒚 = 2𝒙 – 5 for values of 𝒙 from -1 to 3. This is written as the inequality -1 ≤ 𝒙 ≤ 3
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Use the given values for 𝒙 to draw a table for 𝒙 and 𝒚. -1 ≤ 𝒙 ≤ 3 means that the values of 𝒙 start at -1 and go up to 3
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Substitute each value of 𝒙 into the equation 2𝒙 – 5 to find the value of 𝒚. 2𝒙 – 5 means that each value of 𝒙 is multiplied by 2 and then 5 is subtracted to give 𝒚. When 𝒙 = -1, the calculation to find 𝒚 is -1 × 2 – 5. The process is repeated for each value of 𝒙
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Each calculation is evaluated to give 𝒚. When 𝒙 = -1 the calculation -1 × 2 – 5 gives 𝒚 = -7. When 𝒙 = 0 the calculation 0 × 2 – 5 gives 𝒚 = -5. Repeat this for each value of 𝒙
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Use the pairs of values in the table to list the coordinates of the points to be plotted. The coordinates are (-1, -7), (0, -5), (1, -3), (2, -1) and (3, 1).
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Use the coordinates to decide on axes that will take all the values of 𝒙 and 𝒚. The 𝒙-axis needs to include values from -1 to 3. This is shown as the inequality -1 ≤ 𝒙 ≤ 3. The 𝒚-axis needs to include values from -7 to 1. This is shown as the inequality -7 ≤ 𝒚 ≤ 1
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Plot the coordinates for 𝒚 = 2𝒙 – 5. Draw a line through the points and label the line.
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The 𝒚-intercept of the graph is -5. This is the constant (𝒄) in the equation 𝒚 = 2𝒙 – 5. The line crosses the 𝒚-axis at (0, -5).
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The gradient of the line 𝒚 = 2𝒙 – 5 is given by the coefficient of 𝒙 (2). For each one horizontal move to the right, the line moves vertically twice as much up. Eg, one horizontal unit to the right and two vertical units up. Similarly, four horizontal units to the right and eight vertical units up.

Questions

Question 1: Complete the table of values for \(y = 3x + 8\) for values of \(x\) from -2 to 2

Y equals three x plus eight. Minus two is less than or equal to x, which is less than or equal to two. Underneath is a six by two table with values of x on the first row from left to right, minus two, minus one, zero, one, and two. The second row is empty for unknown values of y.

A table of values can also be used to find the coordinates of a line with a negative gradient.

Question 2: Complete the table of values for \(y = 3 – 2x\) for values of \(x\) from -1 to 3

Y equals three minus two x Minus one is less than or equal to x, which is less than or equal to three. Underneath is a six by two table with values of x on the first row from left to right, minus one, zero, one, two and three. The second row is empty for unknown values of y.