3.2 Families of Graphs Section 3.2 Our family of Parent Graphs VERTICAL TRANSLATIONS f ( x) = x2 +1 f x x yy 2 f ( x) = x2 - 3 Above is the graph of x x As you can see, a number added or subtracted from a function will cause a vertical shift or translation in the function. f x x 2 What would f(x) + - 31 look look like? like? (This (This would would mean mean taking taking all all the function values and adding subtracting 3 from them). 1 to them). VERTICAL TRANSLATIONS f ( x) = x + 2 f x x yyy f ( x) = x - 4 Above is the graph of xxx So the graph f(x) + k, where k is any real number is the graph of f(x) but vertically shifted by k. If k is positive it will shift up. If k is negative it will shift down f x x What would f(x) + 2 look like? What would f(x) - 4 look like? HORIZONTAL TRANSLATIONS f ( x) = ( x+ 2) 2 f x x 2 yy y As you can see, a number added or subtracted from the x will cause 2 f ( x) = ( x -1) a horizontal shift or translation in the function but opposite way of 2 the sign of the Above is the graph of f x x number. What would f(x+2) look like? like? (This (Thiswould would mean meantaking taking all all the the xx f(x-1) look values and adding 2 to 1them putting putting them inthem the function). subtracting frombefore them before in the function). x xx HORIZONTAL TRANSLATIONS f x 1 x 1 f x x 3 y y y 3 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. x xx shift right 3 f x 3 x 3 Above is the graph of 3 f x x What would f(x+1) look like? What would f(x-3) look like? 3 (If you set the stuff in parenthesis = 0 & solve it will tell you how to shift along x axis). x 3 0 x3 So shift along the x-axis by 3 We could have a function that is transformed or translated both vertically AND horizontally. yy y up 3 x xx left 2 Above is the graph of f x x What would the graph of f x ( x 2) 3 look like? DILATION: 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. f ( x) = 2 x 4 f x 4 x yy f x x y Notice for any x on the graph, the new (red) graph (green) graph hashas ay value a y value thatthat is 2 is 4 times as much as the original (blue) graph's y value. x x x Above is the graph of f x x What would 2f(x) look like? What would 4f(x) look like? 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 if the value of a was positive but less than 1? f x x yyy 1 f ( x) = x 2 Notice for any x on the graph, the new (green) (red) graph has a ya y graph has 1/2 as much as the value that is 1/4 original (blue) graph's y value. x xx 1 1 f x x 4 4 Above is the graph of f x x What would 1/2 f(x) look like? What would 1/4 f(x) look like? 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. What if the value of a was negative? f x x Notice any x on the new (red) graph has a y value that is the negative of the original (blue) graph's y value. yy xx f ( x) = - x Above is the graph of f x x What would - f(x) look like? 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) There is one last transformation we want to look at. f ( x) = (-x) f x x 3 3 So the graph f(-x) is a reflection about the y-axis of the graph of f(x). y Notice any x on the new (red) graph has an x value that is the negative of the original (blue) graph's x value. y x x (The new graph is obtained by "flipping“ or reflecting the function over the y-axis) Above is the graph of f x x What would f(-x) look like? (This means we are going to take the negative of x before putting in the function) 3 Summary of Transformations So Far **Do reflections and dilations BEFORE vertical and horizontal translations** If a > 1, then vertical dilation or stretch by a factor of a If 0 < a < 1, then vertical dilation or compression by a factor of a If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a) vertical translation of k a f x h k f(-x) reflection about y-axis horizontal translation of h (opposite sign of number with the x) 1 We know what the graph would look like if it was f x x from our library of functions. y y y y x x x moves up 1 1 1 using transformations Graph f x x2 y reflects about the x -axis moves right 2 x x There is one more Transformation we need to know. Do reflections and dilations BEFORE vertical and horizontal translations If a > 1, then vertical dilation or stretch by a factor of a If 0 < a < 1, then vertical dilation or compression by a factor of a If a < 0, then reflection about the x-axis (as well as being dilated by a factor of a) ( x h) a f k b vertical translation of k horizontal translation of h (opposite sign of number with the x) f(-x) reflection about y-axis horizontal dilation by a factor of b
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