Grade 10 Functions and Graphs

Key concepts

• Linear y = ax + q: gradient a and y-intercept q
• Parabola y = ax² + q: a > 0 opens up, a < 0 opens down; turning point (0; q)
• Hyperbola y = a/x + q: asymptotes x = 0 and y = q
• Exponential y = a·bˣ + q (b > 0, b ≠ 1): horizontal asymptote y = q
• Domain and range in interval or set notation
• Finding the equation of a graph from given points

Worked examples

Write down the range of f(x) = 2x² − 3.

a = 2 > 0, so the parabola opens upwards and its minimum value is q = −3. The range is y ≥ −3.

Write down the asymptotes of g(x) = 4/x + 1.

For y = a/x + q, the asymptotes are x = 0 and y = q. So x = 0 and y = 1.

h(x) = 2ˣ − 4. Find the x-intercept.

Set h(x) = 0: 2ˣ = 4 = 2², so x = 2. The x-intercept is (2; 0). The y-intercept is h(0) = 1 − 4 = −3.

Common mistakes

• Drawing a graph that crosses its asymptote.
• Writing the range of a parabola as x-values instead of y-values.
• Mixing up the effect of a and q.
• Plotting too few points, so a hyperbola or exponential graph has the wrong shape.

Exam tips

• For every sketch, label intercepts, turning points and asymptotes; that is where the marks are.
• Draw asymptotes as dotted lines first, then sketch the curve.
• To find a from a sketch, substitute a point that is not the turning point or an intercept on the asymptote.

Frequently asked questions

Do Grade 10 graphs have horizontal shifts?

No. Grade 10 uses the form y = a·f(x) + q only. Horizontal shifts (p), such as y = a(x + p)² + q, are introduced in Grade 11.

What is an asymptote?

A line that a graph approaches but never touches. In Grade 10, hyperbolas have a vertical and a horizontal asymptote, and exponential graphs have a horizontal asymptote.

Are the trig graphs in Paper 1 or Paper 2?

The sine, cosine and tangent graphs are examined with trigonometry in Paper 2. Paper 1 covers the linear, quadratic, hyperbolic and exponential functions.