Warm up State the domain and range for: If f(x) = x 3 – 2x +5 find f(-2)

Graphs of Functions & Translating functions Objective: To recognize and graph functions and translate graphs.

Identity Functions f(x) =x y always = whatever x is; the domain and range are all real numbers. Symmetric with respect to the origin.

Negation Function f(x) = – x y is always the opposite of x; the domain and range are all real numbers. Symmetric with respect to the origin.

Constant Function f(x) = c In this graph the domain is all real numbers but the range is c. Symmetric with respect to the y-axis. c

Squaring Function f(x) = x 2 The graph of a square function is a parabola. Domain is all real numbers, range is y 0 Symmetric with respect to the y-axis.

Square Root Function f(x)= This function is only defined for x ≥ 0, and the range is always y ≥0. The graph is not symmetric.

Absolute Value Function f(x) =|x| The domain is all real numbers. The range is y ≥ 0

Cubic Functions f(x)=x 3 The domain and range are all real numbers Symmetric with respect to the origin.

Translations and reflections of graphs Understanding how the parts of the function affect the graph will make it easier to graph the function without actually plotting individual point.

Shifting Left/Right Shift Inside the parentheses (the function/equation) Opposite sign Example y=|x+2| shift 2 left Up/Down Shift Outside the parentheses (the function/equation) Same sign Example y=|x|+2 shifts up 2

Reflecting Over the x-axis Negative sign outside parentheses Example y=-|x|

Stretching and Shrinking Graphs Numbers larger than 1stretch the graph vertically. Numbers less than 1 shrink the graph vertically.

VERTICAL TRANSLATIONS Above is the graph of What would f(x) + 1 look like? (This would mean taking all the function values and adding 1 to them). What would f(x) - 3 look like? (This would mean taking all the function values and subtracting 3 from them). As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function.

Above is the graph of What would f(x+2) look like? (This would mean taking all the x values and adding 2 to them before putting them in the function). As you can see, a number added or subtracted from the x will cause a horizontal shift or translation in the function but opposite way of the sign of the number. HORIZONTAL TRANSLATIONS What would f(x-1) look like? (This would mean taking all the x values and subtracting 1 from them before putting them in the function).

HORIZONTAL TRANSLATIONS Above is the graph of What would f(x+1) look like? So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis). What would f(x-3) look like? So the graph f(x-h), where h is any real number is the graph of f(x) but horizontally shifted by h. Notice the negative. (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis). So shift along the x-axis by 3 shift right 3

and If we multiply a function by a non-zero real number it has the effect of either stretching or compressing the function because it causes the function value (the y value) to be multiplied by that number. Let's try some functions multiplied by non-zero real numbers to see this. DILATION:

Above is the graph of So the graph a f(x), where a is any real number GREATER THAN 1, is the graph of f(x) but vertically stretched or dilated by a factor of a. What would 2f(x) look like? What would 4f(x) look like? Notice for any x on the graph, the new (red) graph has a y value that is 2 times as much as the original (blue) graph's y value. Notice for any x on the graph, the new (green) graph has a y value that is 4 times as much as the original (blue) graph's y value.

Above is the graph of So the graph a f(x), where a is 0 < a < 1, is the graph of f(x) but vertically compressed or dilated by a factor of a. Notice for any x on the graph, the new (red) graph has a y value that is 1/2 as much as the original (blue) graph's y value. Notice for any x on the graph, the new (green) graph has a y value that is 1/4 as much as the original (blue) graph's y value. What if the value of a was positive but less than 1? What would 1/4 f(x) look like? What would 1/2 f(x) look like?

Above is the graph of So the graph - f(x) is a reflection about the x-axis of the graph of f(x). (The new graph is obtained by "flipping“ or reflecting the function over the x-axis) What if the value of a was negative? What would - f(x) look like? Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value.

Representing Absolute Value The standard form of an Absolute Value Function is:  y = a|x – h| + k The variables a, h and k tell you certain things about this graph.

Graphing Absolute Value Let’s start with a graph The vertex of this graph is at (0,0) and each of the legs has a slope of 1

Graphing Absolute Value Let’s change the slope. Y = 3|x| As slope (the number in front of x) gets larger, the lines get steeper.

Graphing Absolute Value Y = 1/3|x| As slope gets smaller, the lines get shallower.

Graphing Absolute Value Y = -1|x| Changing our value for “a” to a negative has changed our “V” from opening up to opening down.

Graphing Absolute Value Now let’s look at the “h” value. y = |x| Y = |x – 3| This graph has moved right 3 units.

Graphing Absolute Value What do you think the equation of this graph is? y = │ x+2 │

Graphing Absolute Value Now let’s look at the “k” value. y = |x| y = |x| + 4

Graphing Absolute Value What is the equation of this graph?

Graphing Absolute Value So now we know: The value for “a” gives width of graph (wide or narrow) and opening up or down, The value for “h” moves the vertex right or left, The value for “k” moves the vertex up or down

Graphing Absolute Value Based on all this, describe this graph:  y = 3|x – 1| +2

Sources Hotmath.com home.apu.edu/~smccathern/.../slides2_4G raphicalTransformations.pp... seniormath.wikispaces.com/file/.../Shifting +and+Translating+Review....