Curve Sketching Learning Outcomes  Make tables and draw the graphs of various equations to include: Linear Functions Quadratic Functions Cubic Functions Exponential Functions Reciprocal Functions Trig Functions

 Match the correct equations with the correct graph  Solve equations graphically  Interpret the effect of transformation on functions to include: y=f(x+a) applied to f(x) y=f(ax) applied to f(x) y=f(x)+a applied to f(x) y=af(x) applied to f(x)

Curve Sketching Linear Graphs Linear means y = m x + c where m = gradient c = intercept ( y axis) *NB coefficient of y must be 1 (0, -2) (-0.5, 0) y = -4 x – 2 y = 3 x + 2 (2, 0) (- 2 / 3, 0)

Curve Sketching Trig Graphs y = sin x y = cos x y = tan x

Curve Sketching Quadratic and Cubic Graphs Quadratic: y = a x 2 + b x + c General shape for a ≥ 0 General shape for a ≤ 0 Cubic: y = a x 3 + b x 2 + c x + d General shape for a ≥ 0 General shape for a ≤ 0

Curve Sketching Reciprocal and Exponential Graphs Reciprocal curves: y = 1 / x Exponential curves: y = 2 x (a x )

Curve Sketching Curve transformations Translate along x axis – using y = x 2 i)Draw y = ( x + 1) 2 - translation 1 to the left ii)Draw y = ( x - 1) 2 - translation 1 to the right Translate along y axis – using y = x 2 i)Draw y = x translation 1 upwards ii)Draw y = x translation 1 downwards

Curve Sketching Curve transformations Stretch along y axis – using y = sin x sin x i)Draw y = sin 2 x ii)Draw y = sin x /2

Curve Sketching Curve transformations Stretch along y axis y = f(x) → y = af(x) i)Draw y = cos x ii)Draw y = 2cos x i)Draw y = 2 x 2 ii)Draw y = ½ x 2

Curve Sketching Summary TypeNotationExampleEffect Translation i) Along x axis y = F( x + a) y = sin ( x + 90) Sine wave moves 90 ° to the left ii) Along y axis y = F( x ) + a y = x 3 -1 x 3 moves 1 unit down OY Stretch i) Along x axis y = F(a x ) y = sin 2 x Compression To make 2 sine waves ii) Along y axis y = aF( x ) y = 3cos x Stretch scale factor = 3

Curve Sketching Additional Notes

Curve Sketching Learning Outcomes: At the end of the topic I will be able to Can Revise Do Further       Make tables and draw the graphs of various equations to include: Linear Functions Quadratic Functions Cubic Functions Exponential Functions Reciprocal Functions Trig Functions     

        Match the correct equations with the correct graph  Solve equations graphically  Interpret the effect of transformation on functions to include: y=f(x+a) applied to f(x) y=f(ax) applied to f(x) y=f(x)+a applied to f(x) y=af(x) applied to f(x)  