[MUSIC] A linear function takes the general form of y = ax + b, where y is the dependent variable, x is the independent variable. b is the intercept, and a is the gradient. a and b are constants. The graph of the equation is a straight line. If f is the function that assigns y to x, then f(x) = ax + b, and is called a linear function. Let's take an arbitrary value of x, and then we have f(x+1)- f(x) = a times x plus 1 plus b minus ax minus b, equals to a. This shows the measures of the change of y due to the change in x, which increases by one unit. And this is why the number a represents the slope of the line, which describes our function. This is called the gradient, and it's calculated as delta y over delta x, equal to a. This is the variation of y due to the variation of x. If the slope a is positive, the line is upwards to the right. The larger the value of a, the steeper the line. If a is negative, then the line will be downward to the right. And the absolute value of a measures the steepness of the line. If a is equal to 0, the steepness is 0, because the line is horizontal. Let's go through an example. Given the function y=3x+2, if x is equal to 1, it means that y is equal to 3 times 1 plus 2, equal to 5. When x is equal to 2, this means that y = 3 (1) + 5 = 8. Now, the variation of y, which is 8 minus 5 divided by the variation of x, which is 2 minus 1, gives the gradient, which is a equal to 3, which is positive. Then we have delta y over delta x equal to 8 minus 5 over 2 minus 1, which is equal to 3. If we plot the points (1,5) and (2,8), we will have a straight line. And the line is upward to the right. Let's go through another example. If the function is y = 2-3x, if x = 1, it means that y = 2- 3 (1), which is equal to minus 1. When x=2, it means that y= 2-3(2)=-4. Now, the variation of y, which is -4- (-1) divided by the variation of x, which is 2-1 gives the gradient a = -3, which is negative. If we plot the points 1 and -1 and 2 and -4, we have a straight line which is upward to the left. [MUSIC]