0580 · 2.11Sketching curves

Graph features and linear sketches

0:00An equation can tell you the shape of a graph before you calculate a long table of values. Its structure reveals where the graph crosses the axes, turns around, or approaches a line. Sketching brings these clues together into one picture. By the end, you will recognise five families of graphs and explain why their shapes make sense. A graph shows points whose coordinates satisfy an equation. The horizontal coordinate is x, and the vertical coordinate is y.

0:25Substituting a value of x gives the corresponding value of y, when it exists. A sketch is not a carefully scaled plot, but its shape and important features must still be correct. An intercept is where a graph meets an axis. At an x intercept, y = 0, so we find it by solving the equation with y set to zero. The corresponding x value is called a root. At a y intercept, x = 0, so substitution gives its height.

0:50A turning point is where a graph changes from decreasing to increasing, or from increasing to decreasing. A minimum is the bottom of a local dip, and a maximum is the top of a local hill. Symmetry means that a reflection or rotation leaves the graph looking unchanged. A linear graph is a straight line because equal changes in x produce equal changes in y. For y = 2x + 1, increasing x by one increases y by two.

1:16The gradient is therefore two. The added one places the y intercept at the point with x equal to zero and y equal to one. The syllabus also writes linear equations as a times x plus b times y equals c. If b is not zero, rearranging gives y in terms of x. A positive gradient rises from left to right, while a negative gradient falls. A zero gradient gives a horizontal line, and an equation fixing x gives a vertical line.

1:41For our first exam-style example, sketch 2x + 3y = 6. Set x equal to zero to obtain 3y = 6. Dividing by three gives y = 2. This places the y intercept at the point with x equal to zero and y equal to two. Next, set y equal to zero, leaving 2x = 6. Dividing by two gives x = 3. Mark the point with x equal to three and y equal to zero.

2:06Draw a straight line through both intercepts, extending beyond them because these points are not endpoints.

Quadratic graphs and turning points

2:12A quadratic contains an x² term as its highest power. Its graph is a parabola, which has a smooth bowl shape or an upside-down bowl shape. Squaring opposite numbers gives the same result, so y = x² is symmetric about the y axis. Its minimum occurs at the origin because a square cannot be negative. For y equals a times x squared plus b times x plus c, the sign of a determines the opening.

2:36A positive value gives a minimum, while a negative value gives a maximum. The value of c gives the y intercept. The other terms can move the turning point away from the origin. Our second example is y = x² − 6x + 5. To find its turning point, complete the square. Half of negative six is negative three, so start with the square of x − 3. Expanding that square gives x² − 6x + 9.

3:03Subtract nine to recover x² − 6x. Adding the original five then gives y equals the square of x − 3, minus four. The squared part has its smallest value, zero, when x = 3. Therefore, the minimum has x equal to three and y equal to negative four. This form also explains the symmetry rather than just stating it. Values equally far from three give opposite values of x − 3.

3:27Their squares are equal, so their y values match. The axis of symmetry is therefore the vertical line x = 3, passing through the turning point. To find the roots, set y equal to zero. Factorising gives x minus one multiplied by x minus five equals zero. At least one factor must be zero, giving roots of one and five. Substituting x equal to zero into the original equation gives the y intercept of five.

3:52Draw a smooth upward parabola through these intercepts, with its minimum below the x axis. The roots lie equally far from the symmetry line, which checks our calculations. A quadratic can have two roots, one repeated root, or no real roots. These correspond to crossing the x axis twice, touching it once, or missing it completely.

Cubic graphs from roots

4:11A cubic contains an x³ term as its highest power. For y = x³, negative inputs give negative outputs, while positive inputs give positive outputs. Its left end points downwards and its right end upwards. The curve flattens at the origin, but continues increasing, so that point is not a turning point. In y equals a times x cubed plus b, the added b moves the graph vertically. The point with x equal to zero and y equal to b is its centre of rotational symmetry.

4:41Rotating the graph half a turn about that point leaves its shape unchanged. A negative a reverses the directions of the two ends. Our third example is y = x³ − x² − 2x. Factor out x to obtain x multiplied by x squared minus x minus two. Factorising the quadratic gives x multiplied by x minus two multiplied by x plus one. Setting each factor equal to zero gives roots of negative one, zero and two.

5:09A few extra points help establish the shape between those roots. At x equal to negative two, substitution gives −8 − 4 + 4, which equals negative eight. For x equal to −1/2, we obtain negative one eighth minus 1/4 + 1. This simplifies to five eighths, placing that point above the x axis. At x equal to one, the calculation is 1 − 1 − 2, giving negative two.

5:35Substituting x equal to three gives 27 − 9 − 6, which equals twelve. These values show the graph below, above, below, then above the x axis. Its positive leading coefficient also confirms that the right end rises. Draw a smooth curve crossing all three roots, with a hill between negative one and zero and a dip between zero and two. This cubic has two turning points, unlike the basic x³ graph.

6:00The extra squared and linear terms change the middle section. Exact cubic turning-point calculations belong in the video on differentiation, not this sketching method.

Reciprocal graphs and asymptotes

6:10A reciprocal graph contains x in the denominator, as in y = 1/x. When positive x becomes very large, 1 ÷ x becomes very small. As positive x shrinks towards zero, the quotient grows very large. Negative inputs produce a separate branch, and x equal to zero is forbidden because division by zero is undefined. An asymptote is a line that a graph approaches as it extends indefinitely or approaches an excluded input.

6:36Here, the asymptotes are the vertical line x = 0 and the horizontal line y = 0. For these reciprocal graphs, neither branch reaches an asymptote, so never join the branches across the vertical axis. Our fourth example is y = 2/x, then plus one. The denominator still excludes zero, so the vertical asymptote remains x = 0. As x becomes very large in either direction, 2/x approaches zero. Adding one therefore places the horizontal asymptote at y = 1.

7:06For the root, set zero equal to 2/x, plus one. Subtracting one gives −1 = 2/x. Multiplying by x produces −x = 2, so x = −2. There is no y intercept because the function is undefined when x = 0. At x equal to one, y is 2 ÷ 1, plus one, giving three. For x equal to negative one, y becomes −2 + 1, giving negative one. Draw the right branch above y = 1 and the left branch below it.

7:37The graph has rotational symmetry about the point where its asymptotes meet.

Exponential graphs and sketching checks

7:41An exponential has x in the exponent, rather than as a base being squared or cubed. The syllabus form is y = a × rˣ, plus b. The base r is positive. Increasing x by one multiplies the exponential part by r, rather than adding a constant amount. With a positive coefficient, a base greater than one produces an increasing curve. A base between zero and one gives a falling curve instead.

8:07Negative exponents explain the flattening end, since two to the power of negative x is the reciprocal of two to the power of x. A negative coefficient reflects the curve vertically. Our fifth example is y = 3 × 2ˣ, minus three. The exponential part approaches zero towards the left, giving a horizontal asymptote at y = −3. Substituting x equal to zero gives three times one minus three, which equals zero.

8:31The origin is therefore both an x intercept and a y intercept. At x equal to one, y becomes three times two minus three, giving three. Using x equal to two gives three times four minus three, which equals nine. For x equal to negative one, y is three times one half minus three, giving −3/2. Sketch a rising curve through these points, always above its asymptote. Unlike the reciprocal example, this exponential is defined at every real value of x and has no vertical asymptote.

9:03It also has no turning point because it keeps increasing. If r = 1, the exponential expression becomes constant instead. Similarly, a zero leading coefficient can remove the characteristic shape, so always check the actual equation. The most common sketching errors come from drawing a familiar shape before checking the equation. A quadratic does not always have two roots, and a cubic does not always have two turning points. A flattened section is not automatically a turning point.

9:29Label asymptotes separately from the curve, and distinguish their equations from intercept coordinates. In an examination, recognise means identify the function family from its equation or shape. Sketch means show the correct overall shape and the important labelled features. Interpret means explain what the graph tells you, such as its roots or minimum value. Use the equation to justify each feature rather than relying on appearance alone. For a short interpretation question, return to our quadratic and state its minimum value and where y is negative.

9:59Its completed-square form gives a minimum value of negative four at x equal to three. Its roots are one and five. The curve lies below the x axis when x is greater than one and less than five. These methods must work without a calculator on Paper two, and they also apply on calculator Paper four. Completing the square, factorising and finding simple intercepts should be secure by hand. Keep exact values where possible, and do not round during a calculation.

10:25When a numerical answer needs rounding, usually give three significant figures unless the question instructs otherwise. A successful sketch combines the function family with evidence from the equation. Find intercepts, identify turning points and symmetry, then consider the behaviour at the ends. Mark any vertical or horizontal asymptotes before drawing the curve. The result is not just a remembered shape, but a picture whose important features you can explain.