0580 · 2.10Graphs of functions

Tables, points and graph scales

0:00Two different rules can produce the same answer for particular inputs. A graph makes those shared answers visible as intersection points. It can also show where an expression equals zero, or how a quantity changes over time. By the end, you will connect algebraic rules, tables of values and graphs, then use those connections to solve problems. A function gives an output for each allowed input. Here, x usually represents the input and y represents the output.

0:26A coordinate pair records both values together. For example, the point with x equal to two and y equal to four shows that an input of two produces four. To construct a table of values, substitute each chosen input into the rule. Follow the order of operations, calculating powers before adding or subtracting. Put negative inputs in brackets when writing your working. This matters because −2² gives four, whereas the negative of 2² gives negative four.

0:54Every plotted point represents one successful substitution. The complete graph includes the points between your selected inputs, not just those listed in the table. That is why a curved function normally needs a smooth curve rather than straight segments. A table supplies landmarks, while the rule determines the shape between them. Choose scales that fit the requested range and use a substantial part of the grid. Label both axes, and keep equal numerical intervals equally spaced.

1:20The horizontal and vertical scales do not have to match each other. Plot accurately before drawing the curve, because an inaccurate graph produces unreliable solutions.

Power graphs and domains

1:30Many graphs in this topic come from powers of x. The basic rule is y = a × x to the power n. The coefficient a multiplies every output of the power. Start by understanding the shape when that coefficient is one, then use the actual rule to calculate the required values. The power zero produces a constant output of one, for any input other than zero. A constant function such as y = 3 is drawn as a horizontal line.

1:57The power one gives y = x, which is a straight line through the origin. For this rule, every increase of one in the input increases the output by one. Squaring gives equal outputs for opposite inputs because a negative number multiplied by itself is positive. The graph of y = x² therefore has matching left and right sides. Cubing preserves the sign of the input. Its graph passes from negative outputs through the origin to positive outputs, forming a rising curve.

2:24Negative powers mean reciprocals, not negative outputs. The powers negative one and negative two give 1 ÷ x, and 1 ÷ x². Neither expression allows x equal to zero, because division by zero is undefined. Their graphs therefore have separate branches rather than a curve joined across zero. For 1 ÷ x, a negative input gives a negative output. Squaring in the denominator, however, makes every output of 1 ÷ x² positive.

2:54Close to zero, the denominator becomes very small, making the output large in magnitude. Far from zero, the denominator grows, so the output approaches zero. The power 1/2 means square root. Its real number graph starts at the origin and exists only for x greater than or equal to zero. The power −1/2 means 1 ÷ √x. That rule needs x strictly greater than zero, because its denominator cannot be zero.

3:22An allowed set of inputs is called the domain. Check the domain before completing a table, especially when the rule contains a denominator or square root. Never replace an undefined value with zero. An undefined input means there is no point to plot at that input. The coefficient changes the outputs without changing which inputs are allowed, as long as it is not zero. For y = 2x², every output is twice the corresponding output of x².

3:47A negative coefficient reverses the signs of the outputs. You can recognise these effects by comparing calculated points rather than memorising a new graph. A rule can also add up to three power terms. Consider y equals two x plus three divided by x squared. At negative one, the squared denominator is one, giving −2 + 3, which equals one. At positive one, the output is 2 + 3, which equals five.

4:13At zero, the expression remains undefined. Adding terms can produce a shape unlike any single basic power graph. Calculate the whole expression at each input, rather than drawing its separate terms as separate answers. Keep any excluded inputs excluded from the final graph. In this example, you must not join the branches across zero.

Worked example: roots from graphs

4:33Our first worked example asks us to construct a table for y = x³ + x − 4. Use inputs from zero to two in steps of zero point five. At zero, substitution gives 0 + 0 − 4, so the output is negative four. For zero point five, the cube is zero point one two five. Adding zero point five and subtracting four gives negative three point three seven five.

4:57At one, the expression becomes 1 + 1 − 4, giving negative two. For one point five, the cube is three point three seven five. Adding one point five and subtracting four gives zero point eight seven five. At two, the cube is eight. The full calculation is 8 + 2 − 4, giving six. These values provide five points for the graph. Plot each input horizontally and its corresponding output vertically, then draw a smooth curve through them.

5:26The question now asks us to solve x³ + x − 4 = 0 graphically. On our graph, the expression is represented by y. We therefore need the point where y = 0. That point lies on the horizontal axis, and its input is approximately one point three eight. A root is a value of x for which the expression equals zero. Graphically, roots are the horizontal coordinates where the graph meets the x axis.

5:51Read the input, not the vertical coordinate, which is already zero. A graphical result is usually an estimate, with accuracy limited by the scale and quality of your drawing. The same graph can solve equations with other outputs. To solve x³ + x − 4 = 2, draw the horizontal line y = 2. Find its intersection with the curve, then read the horizontal coordinate. This works because both graphs have the same output at that input.

Worked example: graph intersections

6:18Our second worked example is to solve x² = 2x + 3 graphically. Draw the curve y = x² and the line y = 2x + 3. For inputs negative two and negative one, squaring gives four and one. At zero, the curve passes through the origin. The inputs one and two give squared outputs of one and four. At three, the output is nine. Squaring four gives sixteen. Plot these seven points and draw a smooth curve, keeping the matching outputs on either side of zero correctly positioned.

6:49For the straight line, two well separated points are enough to determine its position. At negative two, the calculation gives −4 + 3, producing negative one. When x = 4, the output is 8 + 3, giving eleven. Plot those points and use a ruler to draw the line through them. The graphs cross twice. One crossing has x equal to negative one and y equal to one, and the other has x equal to three and y equal to nine.

7:15So the solutions are x = −1, or x = 3. At each intersection, both rules give the same output. The vertical coordinates are not the solutions because the original equation asks which inputs make the expressions equal. Always inspect the whole requested interval for intersections. One visible crossing does not prove that there is only one solution. A curve might also touch a line without crossing it, and that contact still represents equality.

7:40If the graphs do not meet within the interval, there is no graphical solution there.

Exponential graphs and decay

7:45Our third worked example uses y = 1/4 × 2 to the power x. This is exponential because the input appears in the exponent. At negative one, two to that power equals 1/2. Multiplying by 1/4 gives 1/8, or zero point one two five. At zero, the power of 2 = 1, so the output is zero point two five. For an input of one, multiplying two by 1/4 gives zero point five.

8:13At two, the power is four, producing an output of one. The next input gives 8 × 1/4, which equals two. At four, the power of two is sixteen. Multiplying by 1/4 gives four. Each increase of one in the input doubles the output. This repeated multiplication creates the exponential shape, unlike a straight line, which adds the same amount for each equal input increase. The general exponential form is y = a × b to the power x, then plus c.

8:43Here, b is a positive integer, while a and c are rational numbers. The coefficient a multiplies the power, and c is added afterwards. When x = 0, the output is a + c because b to the power 0 = 1. For a positive coefficient and a base greater than one, the curve rises increasingly steeply. Adding c moves every output up or down by the same amount. With base one, the rule gives a constant, not exponential growth.

9:09A negative coefficient also changes the direction of the graph, so inspect the actual rule. Exponential growth and decay describe repeated multiplication over equal time intervals. Growth uses a multiplier greater than one, while decay uses a positive multiplier less than one. A quantity doubling each hour curves upwards because each increase is larger than the previous increase. A halving quantity decreases by progressively smaller amounts. For a final application, suppose a substance starts with eighty grams and its mass halves every hour.

9:41Its mass equals 80 × 2 to the power negative t, where t is time in hours. At zero hours, the multiplier is one, giving eighty grams. After one hour, multiplication by 1/2 gives forty grams. At two hours, the multiplier is 1/4, so the mass is twenty grams. For three hours, multiplying eighty by 1/8 gives ten grams. Plot time horizontally and mass vertically, including the units. Under this continuous model, a smooth decreasing curve connects the points.

10:13To estimate when the mass reaches thirty grams, draw a horizontal line at that mass. Read the time below its intersection with the curve, giving approximately one point four hours. The quantity approaches zero rather than reaching it after a finite number of halvings. Every halving leaves a positive amount, although eventually it becomes very small.

Graph exam skills and recap

10:32Exam questions may ask you to construct a table, draw a graph, recognise its form, or interpret an intersection. Show substitutions clearly when a missing table entry is requested. Use a smooth curve for curved functions and a ruler for straight lines. Give every solution visible in the specified interval, together with units when the graph represents a practical quantity. Simple substitutions and familiar powers must be manageable without a calculator on Paper two.

10:59Paper four allows a calculator, but you still need correct plotting and interpretation. Keep full calculator values during calculations, then round final answers as the question instructs. With no instruction, use three significant figures. Never claim more accuracy than your graph can show. The most important mistakes come from misunderstanding what a graph represents. Joining across an undefined input invents outputs that do not exist. Reading an intersection’s height instead of its horizontal position answers the wrong question.

11:30Confusing x² with two to the power x ignores whether the input is the base or the exponent. A graph is a picture of all the allowed input and output pairs. Tables give reliable points, while the rule explains the overall shape and any restrictions. Roots occur where the output is zero, and intersections show where two outputs agree. Exponential graphs reveal repeated multiplication, helping you recognise and interpret both growth and decay.