### Some common functions and their graphs

Question Sample Titled 'Some common functions and their graphs'

 (a) Constant functions ${y}$ $={c}$ or ${f{{\left({x}\right)}}}={c}$ , where ${c}$ is a constant The graph is a horizontal line.

${y}={c}$${\left({0},{c}\right)}$${O}$${x}$${y}$

 (b) Linear functions ${y}$ $={a}{x}+{b}$ or ${f{{\left({x}\right)}}}={a}{x}+{b}$, where ${a}$ and ${b}$ are constants and ${a}\ne{0}$ The graph is a straight line.

${O}$${y}={a}{x}+{b}$${x}$${y}$

 (c) Quadratic functions ${y}$ $={a}{x}^{{2}}+{b}{x}+{c}$ or ${f{{\left({x}\right)}}}={a}{x}^{{2}}+{b}{x}+{c}$, where ${a}$ , ${b}$ and ${c}$ are constants and ${a}\ne{0}$ The graph is a parabola.

${O}$${y}={a}{x}^{{2}}+{b}{x}+{c}$${x}$${y}$

 Example The figure shows the graph of ${y}$ $={a}{x}+{b}$ .

${x}$${y}$${1}$${2}$${y}={a}{x}+{b}$

 (a) Write down the ${x}$-intercept and the ${y}$-intercept of the graph. (b) Find the values of ${a}$ and ${b}$ .

Solution

 (a) ∵ The graph cuts the ${x}$-axis and the ${y}$-axis at ${\left({2},{0}\right)}$ and ${\left({0},{1}\right)}$ respectively. ∴ ${x}$-intercept $={2}$ and ${y}$-intercept $={1}$ (b) ∵ The graph passes through ${\left({0},{1}\right)}$ . ∴ By substituting ${\left({0},{1}\right)}$ into ${y}={a}{x}+{b}$ , we have  ${1}$ $={a}{\left({0}\right)}+{b}$ ${b}$ $={1}$ ∵ The graph passes through ${\left({2},{0}\right)}$ . ∴ By substituting ${\left({2},{0}\right)}$ and ${b}={1}$ into ${y}={a}{x}+{b}$ , we have ${0}$ $={a}{\left({2}\right)}+{1}$ ${a}$ $=-\dfrac{{1}}{{2}}$ ∴   ${a}=-\dfrac{{1}}{{2}}$ and ${b}={1}$ .

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