Unit 6

Lesson 8

Do Now

Find the following values for f(x) and g(x):

f(x) = x2 g(x) = 2x2

f(1); f(2); f(0); f(-1); f(-2)g(1); g(2); g(0); g(-1); g(-2)

HW & Objectives

• Be able to reflect graphs in the x and y axes• Be able to shrink or stretch graphs• Be able to recognize odd and even functions

HW: Read p. 249 Do p. 251: 6, 9, 12-28 even, 36, 38, 49, 51, 53, 65, 67, 69

Review

To shift any function, f(x), 3 units up, how does the equation of the function change?Add 3: f(x) + 3To shift any function, f(x), 4 units to the left, how does the equation of the function change?Replace x with x + 4: f(x + 4)

Review

If f(x) = x2 , then vertex is at (0, 0). Where would the vertex be for:

a) g(x) = (x + 5)2 – 6(-5, -6)

b) h(x) = 2(x – 1)2 + 8(1, 8)

New: Reflecting in x and y axes

Go to fooplot and graph f(x) = x^2 and –x^2; how do the graphs compare? Do the same with g(x) = 4x^2 and -4x^2; how do the graphs compare?The graphs are the same except one opens up and one opens down. They are the same except they are reflected over the x-axisf(x) and –f(x) are the same graphs reflected in the x-axis

New: Reflecting in x and y axes

Reflecting in the y-axis occurs when x is replaced in a function with –xFor example, on fooplot, graph f(x) = x^3and g(x) = (-x)^3How do the graphs compare?Try (2x)^(1/2) and (-2x)^(1/2)How do the graphs compare?

New: Stretching and Shrinking

Using fooplot, compare: f(x) = x^3; g(x) = 5(x^3); and 1/5(x^3)These are examples of vertical stretching and shrinkingWhile nothing changes with the x-values, all the f(x) values are changed by a factor of 5 (stretch) or a factor of 1/5 (shrink)

New: Stretching and Shrinking

To stretch or shrink f(x) = x^3 horizontally, replace x with (5x) to shrink it, and (1/5)x to stretch itTry this on fooplot; compare to the graphs from the previous slide alsoNotice that replacing x with 5x means point (1,1) is transformed to (1/5, 1) Notice that replacing x with (1/5)x means point (1,1) is transformed to (5,1)

All Together: Vertical and Horizontal shift, stretch, and shrink; x & y axes reflections

Vertical Shift:add c f(x) + c c units upHorizontal Shift:replace x with x + c f(x + c) c units leftVertical Stretch/Shrink:multiply by c cf(x) c > 1, stretch

0 < c < 1, shrinkHorizontal Stretch/Shrink:replace x with cx f(cx) c > 1, shrink by 1/c

0 < c < 1, stretch by 1/cReflect over x-axis: Reflect over y-axisf(x) -f(x) f(x) f(-x)

PracticeStart with f(x) = x^2; point (1,1) is on the graphStretch the graph vertically by a factor of 3; (1,1)

corresponds to:f(x) = 3(x^2); (1,3) vertical coordinate changes

Shrink vertically by 1/3; (1,1) corresponds to:f(x) = (1/3)(x^2); (1, 1/3) vert. coord. changes

Shrink horizontally by 3:f(x) = (3x)^2; (1/3,1) is on the graph horizontal coordinate changes

Stretch horizontally by 1/3:f(x) = ((1/3)x)^2; (9, 1) on graph hor. coord. changes

New: Even and Odd Functions

f(x) = x^2 is an example of an even functionEven functions are symmetrical with respect to

the y axis; also, f(x) = f(-x)

f(x) = x^3 is an example of an odd functionOdd functions are symmetrical around the origin