Plotting Straight-Line Graphs

Read the horizontal coordinate first, then the vertical coordinate.

Left and down from the origin give negative coordinates.

Start at the origin and use the coordinates in the order written.

Replace x with the given number, then multiply before adding.

Keep the negative sign with the input when you multiply.

Subtract 3 from each x-value separately; the letters label table entries.

Part (a)

1. Substitute x=-1: y=-1-3.
2. Subtracting 3 moves 3 more units below -1, so a=-4.

Part (b)

1. Substitute x=0: y=0-3.
2. So b=-3.

Part (c)

1. Substitute x=2: y=2-3.
2. So c=-1.

Calculate the matching y-value, then write the pair in the order (x, y).

1. First find the matching y: y=2\times(-1)+3=-2+3=1.
2. Keep the given x first and the calculated y second.
3. Plot (-1,1).

Trace vertically from the given x-value to the line, then across to the y-axis.

1. Start at x=1 on the horizontal axis and move vertically to the line.
2. Move horizontally from the line to the vertical axis.
3. The height is y=3.

Think about which coordinate stays the same all along the line.

1. Every point on this line has height -2.
2. The x-coordinate can change, but y stays at -2.
3. Its equation is y=-2.

Read the x-value shared by every point on the vertical line.

1. The line is vertical through x=3, so every point on it has first coordinate 3.
2. The question gives the second coordinate as -1.
3. The point is (3,-1).

Check each point against the equation. Keep the incorrect point’s x-coordinate when correcting its y-coordinate.

Part (a)

1. The equation gives heights -5,-3,-1,1,3 at x=-2,-1,0,1,2.
2. D is plotted at (1,2), so D is incorrect.

Part (b)

1. At x=1, y=2\times1-1=1.
2. The corrected point is (1,1).

For (a), trace up and across. For (b), look where the line crosses the x-axis.

Part (a)

1. The horizontal scale increases by 1 per interval. Start at x=3 and move up to the line.
2. The vertical scale increases by 2 per interval. Read across to y=8.

Part (b)

1. A point with y=0 lies on the horizontal axis.
2. The line crosses that axis one unit left of the origin, so x=-1.

Subtract each endpoint’s x-value from 4 to find its height; keep (x, y) in order.

Part (a)

1. Subtract x from both sides: y=4-x.
2. At x=-1, y=4-(-1)=5.
3. The endpoint is (-1,5).

Part (b)

1. Use y=4-x. At x=5, y=4-5=-1.
2. The endpoint is (5,-1).
3. The completed graph is shown below.

For (a), try the column with x = 0 first, then check another column. For (b), use the rule you found.

Part (a)

1. At x=0, the table gives y=-1, so the constant term must be -1.
2. Check y=2x-1 at x=2: 2\times2-1=3. At x=-2: 2\times(-2)-1=-5.
3. All three columns match y=2x-1.

Part (b)

1. Use the matching rule y=2x-1.
2. At x=3, y=2\times3-1=5.

The intersection is one point shared by both lines; read both of its coordinates.

1. Find the crossing of the two lines.
2. Read down from the crossing to get x=2, and across to get y=3.
3. The intersection is (2,3).

For (a), check each column against the equation. For (b), recalculate the incorrect height. For (c), compare the changes between successive heights.

Part (a)

1. Check each input in y=2x+3. The correct heights for x=-2,-1,0,1,2 are -1,1,3,5,7.
2. Only the first column disagrees with the table, so the incorrect entry is at x=-2.

Part (b)

1. At x=-2, y=2\times(-2)+3=-4+3=-1.
2. Keep the input first: the corrected point is (-2,-1).

Part (c)

1. The inputs increase by 1 each time. For y=2x+3, every matching height must therefore increase by 2.
2. The original heights are -7,1,3,5,7: their changes are 8,2,2,2. The first segment would have a different slope from the others, making a kink.
3. The equation gives a straight line, so correct the first point before drawing it.
4. The completed graph is shown below.

For (a), x = 1 is halfway between the given x-values. For (b), repeat the same change from A to B.

Part (a)

1. From A to B, x increases by 2 and y increases by 1-(-2)=3.
2. The value x=1 is halfway from 0 to 2, so its height is halfway from -2 to 1.
3. Half of the rise is 3\div2=1.5, giving y=-2+1.5=-0.5.

Part (b)

1. Going from x=2 to x=4 is another increase of 2.
2. A straight line has a steady change, so y increases by another 3: 1+3=4.
3. The height is y=4.
4. The completed graph is shown below.

Use the crossing, not either axis intercept. Each small interval is 0.2 units.

1. Read the crossing carefully. Its horizontal coordinate is between 1.6 and 1.8, a little closer to 1.6: about 1.7 to 1 decimal place.
2. Its vertical coordinate is between 2.2 and 2.4, a little closer to 2.4: about 2.3 to 1 decimal place.
3. The estimated intersection is (1.7,2.3).
4. As a check, equal heights give 2x-1=4-x, so 3x=5 and x=5/3. The matching height is 7/3. These round to the same coordinates.

For (a) and (b), read the coordinates of the crossing. For (c) and (d), compare the two heights at 5 hours.

Part (a)

1. Equal cost occurs where the lines cross.
2. Read the horizontal coordinate of the crossing: x=3.
3. The charges are equal after 3 hours.

Part (b)

1. Read the height of the crossing: £10.
2. Both companies charge £10 for 3 hours.

Part (c)

1. At 5 hours, line A is lower than line B, so Company A is cheaper.

Part (d)

1. At 5 hours, read £14 for Company A and £16 for Company B.
2. The difference is 16-14=2, so Company A is £2 cheaper.

For (a), put both coordinates of the given point into the equation. Then use the completed rule separately at each endpoint.

Part (a)

1. The point (1,5) means x=1 and y=5.
2. Substitute them: 5=2\times1+c=2+c.
3. Subtract 2 to get c=3, so the line is y=2x+3.

Part (b)

1. Use the rule from part (a): y=2x+3.
2. At x=-3, y=2\times(-3)+3=-6+3=-3.
3. The endpoint is (-3,-3).

Part (c)

1. Use y=2x+3. At x=2, y=2\times2+3=7.
2. The endpoint is (2,7). The straight segment through the endpoints also passes through the given point (1,5).
3. The completed graph is shown below.