Algebra, Year 10: function notation and straight lines

8; 0; −4

提示Replace x by the number in the brackets, work out the inside first, then double.

为什么f(1) = 2 × (1 + 3) = 2 × 4. f(−3) = 2 × 0. f(−5) = 2 × (−2). The number in the brackets after f is the input; the rule on the right says what to do with it.

3

提示Write the rule equal to zero and solve it like any other equation.

为什么f(x) = 0 means 2x − 6 = 0, so 2x = 6 and x = 3. Check: f(3) = 2 × 3 − 6 = 0. Here the output is given and the input has to be found, the reverse of working out f(3).

When the input is 4, the output is 11

提示One of the two numbers goes into the rule and the other comes out.

为什么The number in the brackets is what is fed in, and the number after the equals sign is what the function returns. On the graph of y = f(x), the statement says the point (4, 11) lies on the curve.

4a² + 4a

提示Put the whole of the new input, in brackets, wherever x appears in the rule.

为什么f(2a) = (2a)² + 2(2a). Squaring the bracket squares the 2 as well as the a, giving 4a², and 2 × 2a = 4a. The input does not have to be a number; any expression can be fed in.

f(x)

提示It is written with the name of the function followed by the input in brackets.

为什么It is read "f of x". Writing f(x) = 3x + 2 names the rule f and says what it does to any input x; then f(5) means the output when 5 is fed in, which is 17.

gradient

提示It is the number that multiplies x when the equation is written y = mx + c.

为什么Lines that climb at the same rate stay the same distance apart and never meet. In y = 3x + 1 and y = 3x − 4 both lines go up 3 for every 1 across, so they are parallel.

y = 3x + 5

提示In each equation, find the number that multiplies x.

为什么The line y = 3x − 2 has gradient 3. The gradients of the three choices are −2, 3 and 1/3, so only y = 3x + 5 matches. The matching −2 in y = x/3 − 2 is the intercept, which has nothing to do with being parallel.

They have the same gradient, so one is always exactly 3 units above the other

提示Compare the heights of the two lines at any one x-value, then at another.

为什么At every x-value the second line is 4 − 1 = 3 higher than the first: at x = 0 the heights are 1 and 4, at x = 5 they are 11 and 14. The gap never closes, so there is no crossing point.

y = 4x + 5

提示A parallel line keeps one of the two numbers in the equation; the given point lets you find the other.

为什么Parallel means the same gradient, so the new line is y = 4x + c. Substituting (1, 9) gives 9 = 4 + c, so c = 5. Check: 4 × 1 + 5 = 9.

−3

提示Rearrange the equation so that it starts "y =" before reading anything off.

为什么Subtracting 3x from both sides gives y = −3x + 4, so the gradient is −3. A parallel line has the same gradient. The gradient can only be read from an equation once y is the subject.

y = 3x + 2

提示Find the steepness from the two points first, then use one of them to find where the line crosses the y-axis.

为什么The gradient is (11 − 5) ÷ (3 − 1) = 3, so the line is y = 3x + c. Putting in (1, 5) gives 5 = 3 + c, so c = 2. Check with the other point: 3 × 3 + 2 = 11.

y = 2x − 5

提示Start from y = 2x + c and use the point to find the missing number.

为什么Substituting x = 3 and y = 1 into y = 2x + c gives 1 = 6 + c, so c = −5. The intercept is negative because, going back 3 units from the point, the line drops 6 units from a height of 1.

−1/2; 8

提示From the first point to the second, see how far the line moves across and how far it moves down.

为什么The change in y is 5 − 7 = −2 and the change in x is 6 − 2 = 4, so the gradient is −2/4 = −1/2. Then y = −x/2 + c, and (2, 7) gives 7 = −1 + c, so c = 8. Check with (6, 5): −3 + 8 = 5.

y = 4

提示Compare the two y-coordinates before calculating anything.

为什么Both points have a y-coordinate of 4, so the line is horizontal: its gradient is (4 − 4) ÷ (5 − (−2)) = 0. In y = mx + c that leaves y = 0x + 4, which is just y = 4.

point

提示You need an x-value and a y-value that you are sure fit the equation.

为什么For a line of gradient 3 through (2, 10): 10 = 3 × 2 + c, so c = 4. Any place on the line will do, and using a second one afterwards is a good check.