The graph of f is given. Sketch the graphs of the following functions. (a) y=f(x-2) (b)

The graph of f is given. Sketch the graphs of the following...

Key Concepts

Horizontal Translation

A horizontal translation shifts the graph left or right without changing its shape. This occurs when the input variable x in the function is replaced by x minus a constant. For example, f(x-2) moves the graph of f two units to the right, because the effect is equivalent to adjusting every x-coordinate by adding 2.

Vertical Translation

A vertical translation shifts the graph up or down without altering its shape. This is achieved by adding or subtracting a constant from the function value. For instance, f(x) - 2 shifts the entire graph of f downward by two units, while adding a constant such as in -f(x) + 3 shifts it upward after any other transformation.

Vertical Scaling

Vertical scaling stretches or compresses the graph in the vertical direction, changing the distance between points and the x-axis. Multiplying the function by a constant greater than 1 stretches it, whereas a constant between 0 and 1 compresses it. For example, 2f(x) doubles the height of the graph, while (1/2)f(x) compresses it vertically by a factor of one half.

Reflection

Reflections flip the graph over a specified axis. Replacing f(x) with -f(x) causes a reflection over the x-axis, which inverts the vertical values of the graph. Similarly, replacing x with -x as in f(-x) reflects the graph over the y-axis, effectively mirroring the graph horizontally.

Composite Transformations

Composite transformations involve applying multiple graph alterations sequentially to the original function. These can combine translations, scalings, and reflections. An example is (1/2)f(x-1), where the graph of f is first shifted horizontally by one unit and then vertically compressed by a factor of one half. Understanding how each individual transformation affects the graph is key to accurately sketching the composite function.

Transcript

00:01 We have problem number 63 in which we have been provided with a graph as shown and we need to draw the graph of graphs of the functions that is a start y equal to f of x minus 2 so this is the shifting of the graph two units to the right two in to the right so this will move two units that is it will come here because one box is equal to two units units blue color graph is a graph of y equal to f x minus 2 this is the graph of a now part b y equal to f x minus 2 so this graph fx graph fx or this is fx minus 2 fx minus 2 f of x separately and minus 2 so the graph of effect that is a will be shifted two units to the two units to the vertically downward so this is zero this will be minus two so our graph will be this green color graph is the graph of b now part c is y equal to y equal to two of f x so this is vertically stretching by a factor by a factor to keeping x coordinate constant so if this is four this should be eight so red colored graph is a graph of c graph of d is the formula for d is formula for d is y equal to minus of f x plus three so first of all we should take reflection with respect to x -axis and then move three units up so reflection with respect to to xx x x x will be here and it will be moved two units up so reflection will look like this two units up will look like so this is the graph of part d part is y equal to f of minus x y equal to f of minus x so it is reflection on y x reflection on y x will look like this so this is the graph of part e that is reflection now part f y equal to half of f x minus one so we should first shift this graph f x one unit right it will come here and then then shrink the graph by half of value half vertically by a factor of one by two along xx is along xasory so the graph will look like okay we will just draw another graph for part f so we should take the graph paper yes so we have been provided with the graph of this is y x the graph of y equal to fx was four one two three four it was like this okay now part of what we need to shift this graph one unit to the right.

05:11 So one unit to the right will be here and then shrink by half.

05:21 So this y coordinate equal to four...

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