Question

Sketch the graph of the first function by plotting points if necessary. Then use transformation(s) to obtain the graph of the second function. \(y=|x|, \quad y=|2 x-1|+1\)

Short answer

Plot the graph of \(y = |x|\) using chosen values of \(x\) and corresponding \(y\) values. The graph is V-shaped and symmetric about the y-axis. Apply the necessary transformations to the second function, \(y=|2x-1|+1\), by horizontally scaling and shifting the points of the first function. Plot these transformed points, and the resulting graph is also V-shaped but horizontally scaled and shifted vertically.

Step-by-step solution

Plot and understand the first function, \(y = |x|\)

To plot the function \(y = |x|\), choose several values of \(x\) and then calculate the corresponding values of \(y\). Choose both positive and negative values of \(x\) to see the graph's behavior. Here are some chosen values and their absolute values: - For \(x = -3\), \(y = |-3| = 3\) - For \(x = -2\), \(y = |-2| = 2\) - For \(x = -1\), \(y = |-1| = 1\) - For \(x = 0\), \(y = |0| = 0\) - For \(x = 1\), \(y = |1| = 1\) - For \(x = 2\), \(y = |2| = 2\) - For \(x = 3\), \(y = |3| = 3\) Now plot these points on a Cartesian plane. We can see that the graph is V-shaped and symmetric about the y-axis.

Identify the transformation and apply it

The second function, \(y = |2x - 1| + 1\), has undergone two transformations. First, the function has been scaled horizontally by a factor of 2 and shifted downwards by 1 unit, resulting in \(y = |2x - 1|\). Second, the graph has been shifted upward by 1 unit. To sketch the graph of the second function, we will apply these transformations to the original function \(y = |x|\).

Apply the scaling and shifting transformations

To apply the transformations, we will substitute the original \(x\) values with new values that result from the transformations. For the horizontal scaling and shift, we will replace \(x\) with \(\frac{1}{2} (x + 1)\). Then, we will add 1 to the resulting absolute value as the vertical shift: $$ y = \left|\frac{1}{2}(x + 1)\right| + 1 $$ Using the same \(x\) values as before, calculate the new function's values: - For \(x = -3\), \(y = \left|\frac{1}{2}(-3 + 1)\right| + 1 = 2\) - For \(x = -2\), \(y = \left|\frac{1}{2}(-2 + 1)\right| + 1 = \frac{3}{2}\) - For \(x = -1\), \(y = \left|\frac{1}{2}(-1 + 1)\right| + 1 = 1\) - For \(x = 0\), \(y = \left|\frac{1}{2}(0 + 1)\right| + 1 = \frac{3}{2}\) - For \(x = 1\), \(y = \left|\frac{1}{2}(1 + 1)\right| + 1 = 2\) - For \(x = 2\), \(y = \left|\frac{1}{2}(2 + 1)\right| + 1 = \frac{5}{2}\) - For \(x = 3\), \(y = \left|\frac{1}{2}(3 + 1)\right| + 1 = 3\)

Plot the second function

Now plot these new points on the same Cartesian plane as the first function. We can see that the resulting graph is also V-shaped, like the original function. However, the graph is horizontally scaled and shifted vertically.