which one is a linear function?

Linear functions are those where the independent variable x never has an exponent larger than 1. So for example they would not have a var such as 3x2 in them. The linear function on this page is the general way we write the equation of a straight line. It is of the form

y = ax + b Where:

x,y are the coordinates of any point on the line

a is the slope of the line

b is the y-intercept (where the line crosses the y-axis)

The a var is the slope of the line and controls its 'steepness'. A positive value has the slope going up to the right. A negative slope goes down to the right.

The b var is the y intercept - the point where the line crosses the y axis. Adjust the sliders above to vary the values of a and b, and note the effects they have on the graph.

Another form

The more common form of the linear function is written y = mx+b, using m for the slope instead of a. This version is included to be consistent with the quadratic and cubic explorers. If you prefer it the usual way use Linear explorer (mx+b).

Things to try

The simplest case. Y = constant. (y = b)

Click 'zero' below each slider

Since a and b are both set to zero, this is the graph of the equation y = 0x+0. This simplifies to y = 0 and is of course zero for all values of x. Its graph is therefore a horizontal straight line through the origin.

Now move the rightmost slider for b and let it settle on, say, 5.

This is the graph of the equation y = 0x+5. This simplifies to y = 5 and so the function has the value 5 for all values of x. It is therefore a straight horizontal line through 5 on the y axis. Play with different values of b and observe the result.

Linear equation. (y = ax+b)

Click 'reset'

Click 'zero' under the right b slider.

The value of a is 0.5 and b is zero, so this is the graph of the equation y = 0.5x+0 which simplifies to y = 0.5x. This is a simple linear equation and so is a straight line whose slope is 0.5. That is, y increases by 0.5 every time x increases by one. Since the slope is positive, the line slopes up and to the right. Since b is zero, the y-intercept is zero and the line passes through the origin (0,0). Play with the a slider and observe the results, including negative values.

Click on 'reset' and move the b slider to, say, 8.

The value of a is 0.5 and b is 8, so this is the graph of y = 0.5x+8. The effect of changing b from zero to 8 is that the graph has moved upwards and now passes through 8 on the y axis.

Move both sliders and observe the overall effects of these two coefficients (a and b) working together.

Try it yourself

Press "reset", then "hide details"

Adjust the sliders until you see a line that appeals to you

Estimate the slope and y-intercept of the line and write down the equation for the line

Click on "show details" and see how close you got

Hints

The point where the line crosses the (vertical) y-axis is the y-intercept - b

For each increase of one on the (horizontal) x-axis, how much does the line go up or down?

This is a, the slope of the line

If the line goes down and to the right, the slope (a) will be negative.

Coordinate Geometry

In coordinate geometry, the equation for a straight line is usually written y = mx+b. That is, the letter m is used to indicate the slope. See Equation of a line (coordinate geometry).