Algebra, Year 10: function notation and straight lines

8; 0; −4

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

The whyf(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

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

The whyf(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

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

The whyThe 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

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

The whyf(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)

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

The whyIt 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

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

The whyLines 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

HintIn each equation, find the number that multiplies x.

The whyThe 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

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

The whyAt 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

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

The whyParallel 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

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

The whySubtracting 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

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

The whyThe 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

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

The whySubstituting 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

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

The whyThe 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

HintCompare the two y-coordinates before calculating anything.

The whyBoth 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

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

The whyFor 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.