Create an Equation and Sketch a Graph That is Linear is Continuous

Linear equation graphs

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 \(y = mx + c\)

The graph of a oblique straight line is described using the equation, 𝒚 = 𝒎𝒙 + 𝒄

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.

Label to the left of a graph y equals x. Graph showing the x axis and y axis increasing in units of five and ten and decreasing in units of minus five and minus ten. Intersecting diagonally through the origin point labelled, open brackets, nought, nought close brackets and below is the word origin. The positive plot line rises to the right through two points. The first point is open bracket four, four close bracket. The second point is open bracket six, six close bracket. At the top end of the plot line is y equals x. At the origin point, the plot line representing negative descends diagonally to the left with two points labelled open bracket minus four, minus four, close bracket and open bracket, minus eight, minus eight, close bracket. — 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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Example one. Graph showing the x axis and y axis increasing in units of five and ten and decreasing in units of minus five and minus ten. Intersecting diagonally through the central point labelled, open brackets, nought, nought close brackets, highlighted in orange. The right hand section of the graph highlighted in orange is a point, open brackets eight, eight close brackets. To the bottom left section of the graph highlighted in orange is a point, open bracket minus seven, minus seven close bracket. — 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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Graph showing the x axis and y axis increasing in units of five and ten and decreasing in units of minus five and minus ten. Intersecting diagonally through the origin point is a plot line with points in the negative left hand section of the graph, central point and a point in the right hand positive section. At the top of the plot line is y equals x. — 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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Example two. Graph showing the x axis and y axis increasing in units of five and ten and decreasing in units of minus five and minus ten. To the upper left section of the graph is a point, open bracket, minus seven, seven close bracket and highlighted orange. To the bottom left section point, open bracket, two, minus two close bracket is highlighted orange. Second point further down is open bracket, nine, minus nine close bracket is highlighted orange. — 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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\(m\) 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 value of \(y\) . 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.

Y equals two x minus five. Minus one is less than or equal to x, which is less than or equal to three. — 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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The same equations as the previous image. The inequality, minus one is less than or equal x which is less than or equal to three, is highlighted orange. 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 – these are highlighted orange. The second row is empty for unknown values of y. — 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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The same equations and table as the previous image. In the equation y equals two x minus five, the two is highlighted pink and the minus five is highlighted green. The tables of values is filled with equations to work out y. In the grid space on the second row below minus one is the equation minus one multiplied by two minus five – multiplied by two is highlighted pink and minus five is highlighted green. In the grid space below zero is the equation zero multiplied by two minus five. In the grid space below one is the equation one multiplied by two minus five. In the grid space below two is the equation two multiplied by two minus five. In the grid space below three is the equation three multiplied by two minus five. — 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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The same equations and tables as the previous image. On the table, y is highlighted blue. In the grid space below minus one, underneath the equation minus one multiplied by two minus five, is the y value minus seven – minus seven is highlighted blue. In the grid space below zero, underneath the equation zero multiplied by two minus five, is minus five – minus five is highlighted blue. In the grid space below one, underneath the equation one multiplied by two minus five, is minus three – minus three is highlighted blue. In the grid space below two, underneath the equation two multiplied by two minus five, is minus one – minus one is highlighted blue. In the gird space below three, underneath the equation three multiplied by two minus five, is one – one is highlighted blue. — 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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The same equations and table as the previous image. Open bracket x comma y close bracket is written underneath the table. Open bracket minus one comma minus seven close bracket is underneath the first column– minus one is highlighted orange and minus seven is highlighted blue. Open bracket zero comma minus five close bracket is underneath the second column of the table – zero is highlighted orange and minus five is highlighted blue. Open bracket one comma minus three close bracket is underneath the third column – one is highlighted orange and minus three is highlighted blue. Open bracket two comma minus one close bracket is underneath the fourth column – two is highlighted orange and minus one is highlighted blue. Open bracket three comma one close bracket is underneath the fifth column – three is highlighted orange and one is highlighted blue. — 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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A graph showing the x axis increasing in units of one from minus two to four – this is highlighted orange. The y axis increasing in units of one from minus seven to one – this is highlighted blue. They intersect at zero comma zero. The equation y equals two x minus five is in the top left. The coordinates open bracket minus one comma minus seven closed bracket, open bracket zero comma minus five close bracket, open bracket one comma minus three close bracket, open bracket two comma minus one close bracket, and open bracket three comma one close bracket, are on the right hand side of the graph. The x coordinates are highlighted orange and the y coordinates are highlighted blue. — 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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Same graph as previous image. A line is drawn through the five coordinates listed – it is highlighted orange. It is labelled y equals two x minus five. — Plot the coordinates for 𝒚 = 2𝒙 – 5. Draw a line through the points and label the line.

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Same graph as previous image. Minus five on the y axis is highlighted green. Minus five in the equation y equals two x minus five is highlighted green. — 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 same graph as the previous image. Two in the equation y equals two x minus five is highlighted pink. A dotted arrowed line points right from open bracket minus one comma seven close bracket four units across the graph – this is highlighted pink. The number four is under the line – this is highlighted pink. From the end of this line, another dotted arrowed line extends eight units up the graph – this is highlighted pink. The number eight is next to the line – this is highlighted pink. Another dotted arrowed line points right from open bracket one comma minus three close bracket one unit across the graph – this is highlighted pink. The number one is under the line – this is highlighted pink. From the end of this line, another dotted arrowed line extends two units up the graph – this is highlighted pink. The number two is next to the line – this is highlighted pink. — 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.

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A position on a graph is defined by coordinates ( \(x\) , \(y\) ). When one coordinate is given, the second can be read from the graph.

To find a \(y\) -coordinate from a given \(x\) -coordinate:
1. On the \(x\) -axis, locate the given amount.
2. Draw a vertical line, using a ruler, from the given amount up to the line.
3. Draw a horizontal line, using a ruler, from the line across to the \(y\) -axis.
4. Read the value on the \(y\) -axis.
To find an \(x\) -coordinate from a given \(y\) -coordinate:
1. On the \(y\) -axis, locate the given amount.
2. Draw a horizontal line, using a ruler, from the given amount across to the line.
3. Draw a vertical line, using a ruler, from the line down to the \(x\) -axis.
4. Read the value on the \(x\) -axis.

Example one. A graph showing the x axis increasing from minus five to five and the y axis increasing in units of tens from minus ten to thirty, they intersect at zero comma zero. An oblique line slopes up the graph from left to right labelled y equals five x plus four. To the right of the graph, the equations x equals four and y equals question mark. — Use the graph to find the value of 𝒚 when 𝒙 = 4

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The same graph as the previous image. A dotted line extends from four on the x axis to the oblique line, it is labelled x equals four – this is highlighted orange. — On the 𝒙-axis, locate the given amount (4) and draw a vertical line, using a ruler, up to the line.

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The same graph as the previous image. A second dotted line extends from where the oblique line meets the first dotted line, to the y axis – this is highlighted blue. It is labelled y equals twenty four. To the right of the graph, the equation is completed, y equals twenty four. — Then, draw a horizontal line, using a ruler, from the line across to the 𝒚-axis. Read the value on the 𝒚-axis. When 𝒙 = 4, 𝒚 = 24

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Example two. A graph showing the x axis increasing in units of two from minus eight to twenty. The y axis increasing in units of two from minus two to twelve. They intersect at zero comma zero. An oblique line slopes down the graph from left to right labelled y equals eight minus x divided by two. To the right of the graph, the equations y equals two and x equals question mark. — Use the graph to find the value of 𝒙 when 𝒚 = 2

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The same graph as the previous image. A dotted line extends from two on the y axis to the oblique lie, it is labelled y equals two – this is highlighted blue. — On the 𝒚-axis, locate the given amount (2) and draw a horizontal line, using a ruler, across to the line.

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The same graph as the previous image. A second dotted line extends from where the oblique line meets the first dotted line to the x axis – this is highlighted orange. It is labelled x equals twelve. To the right of the graph, the equation is completed, x equals twelve. — Draw a vertical line, using a ruler, from the line down to the 𝒙-axis. Read the value on the 𝒙-axis. When 𝒚 = 2, 𝒙 = 12

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Practise reading and plotting linear equation graphs with this quiz. You may need a pen and paper to help you.

Linear graphs are commonly used when converting between different units of measurement.

For example, swapping between temperatures in degrees Celsius (°C) and degrees Fahrenheit (°F), exchanging between different currencies, such as pounds and euros, or changing inches into centimetres.

A linear graph could be used when exchanging euros into pounds, two different types of currency.

Giving the correct strength of a drug to a patient is vital.

Linear graphs are useful to pharmacists and scientists in the pharmaceutical industry when working out the correct strength of drugs.

The amount of a drug for a given volume of medicine is critical, both for the medicine to be effective and for the safety of the patient.

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Source: https://www.bbc.co.uk/bitesize/topics/zdbc87h/articles/zkpj2nb

Create an Equation and Sketch a Graph That is Linear is Continuous

Linear equation graphs

Jump to

Key points

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

Examples

Question

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

Example

Questions

Reading 𝒙 and 𝒚 coordinates from a graph

Examples

Question

Practise reading and plotting linear equation graphs

Quiz

Real-life maths

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Create an Equation and Sketch a Graph That is Linear is Continuous

Linear equation graphs

Jump to

Key points

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

Examples

Question

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

Example

Questions

Reading 𝒙 and 𝒚 coordinates from a graph

Examples

Question

Practise reading and plotting linear equation graphs

Quiz

Real-life maths

Related Posts

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