Sketch the graph of each function by transforming the graph of an appropriate function of the fo

step 1 So, in this question we have to draw the four given graphs. So I have this, the x -axis, the y -

Sketch the graph of each function by transforming the graph...

Key Concepts

Horizontal Translation

A horizontal translation shifts the graph of a function left or right without altering its overall shape. This transformation is reflected in functions when x is replaced by (x - h) or (x + h), where h is a constant. The graph shifts to the right by h units when written as (x - h) and to the left when written as (x + h), thereby moving the location of key features such as intercepts without changing the graph's orientation.

Vertical Translation

Vertical translation moves the graph up or down without changing its shape. This is achieved by adding or subtracting a constant outside the function. When a function f(x) is transformed to f(x) + k, the whole graph is shifted upwards by k units if k is positive, and downwards if k is negative. This type of transformation impacts the vertical location of key points including the y-intercept.

Vertical Dilation (Scaling)

Vertical dilation or scaling involves stretching or compressing the graph of a function in the vertical direction. Multiplying the function by a constant factor greater than 1 stretches the graph vertically, making the peaks and valleys more pronounced, while a factor between 0 and 1 compresses it, reducing the vertical distances. This transformation affects the steepness of the graph and the spacing of its features, such as intercepts and extrema.

Reflection

Reflection is a transformation that flips the graph of a function over a specific axis. Reflecting across the x-axis, for instance, changes the sign of the function’s output, turning f(x) into -f(x). This operation reverses the graph vertically, altering the positions of intercepts and the overall orientation of the original graph. Such reflections are key when dealing with functions that may have a mirror image relative to the x-axis.

Intercepts

Intercepts are points where the graph of a function crosses the axes. The x-intercepts are found by setting the function equal to zero and solving for x, while the y-intercept is obtained by evaluating the function at x = 0. Identifying intercepts is an essential part of graphing functions because they provide fixed points that help in anchoring the position and shape of the graph relative to the coordinate axes.

Graph Transformations

Graph transformations involve modifying the basic shape of a function’s graph through operations such as translation, scaling, and reflection. These modifications are applied to a standard or base function to obtain a new function, and they allow a systematic way of understanding how the graph’s features change. In the context of polynomial functions, such as those of the form y = x^n, these transformations help in sketching graphs by shifting, stretching, compressing, or reflecting the original graph.

Transcript

00:01 So, in this question we have to draw the four given graphs.

00:06 So i have this, the x -axis, the y -axis.

00:09 So the px graph is very simple.

00:11 First of all, i will draw that.

00:12 So x equal to minus 3 is the root.

00:15 And using that, i will draw a 5 degree polynomial, which will have this kind of an appearance.

00:22 All right.

00:22 So this is the root.

00:25 It is minus 3 comma 0.

00:28 So this is simple and straightforward.

00:30 Now looking at the other functions, 2x plus 3 whole rest to the power 5 minus 64.

00:37 For this we have to understand i will solve here for qx.

00:41 Qx equal to 0 let me solve it.

00:43 So that means x plus 3 rest to the power 5 is equal to 2 rest to the power 5 i will be getting.

00:50 So x plus 3 is equal to 2 x equal to minus 1.

00:54 So that means x equal to minus 1 will be the root and this graph will be x.

01:00 Exactly the same.

01:02 So px graph multiplied by 2 will also have the same curvature.

01:07 But so here i have the graph of qx drawn separately.

01:15 So it is to be observed that the root of qx is x equal to minus 1 that i have got it already.

01:20 So i'll mark this point as minus 1 comma 0 which is the root and apart from that it is a continuously increasing graph.

01:28 So when i put x equal to minus 3, so this will be minus 3 comma minus 3...

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