Problem 160 - Algebra and Trigonometry Real Mathematics, Real People

Question

Sketch the graph of the function. $$f(x)=|x-2|-8$$

Step-by-Step Solution

Answer

The graph of the function \(f(x) = |x - 2| - 8\) is a 'V'-shaped curve, with the vertex of the 'V' located at the point (2, -8). It mirrors around the vertical line \(x = 2\) and opens upwards.

Step 1: Understand the Basic Function and Transformations

The function \(f(x) = |x|\) is a basic absolute value function. It produces a 'v'-shaped graph centered around the point (0,0). When we make the change to \(f(x) = |x - 2| - 8\), we're modifying the original function by shifting it two units to the right and eight units downward.

Step 2: Determine the Vertex

The highest or lowest point of an absolute value function is called the vertex. In the given function \(f(x) = |x - 2| - 8\), because of the shift to the right, the vertex moves from (0,0) in the basic function to (2, -8). This suggests this absolute value function has a vertex at the point (2, -8).

Step 3: Plot the Vertex

Start by plotting the vertex (2, -8) on a graph. This point represents the lowest point on the graph, where the function hinges and changes direction.

Step 4: Plot Additional Points

As an absolute value function, the two halves of the graph will mirror each other around a vertical line through the vertex. To sketch the graph, select a couple of x-values on either side of the vertex, find corresponding y-values, and mark these points. For example, when \(x = 1\), \(f(x) = |1 - 2| - 8 = -7\), so you can plot point (1, -7). When \(x=3\), \(f(x) = |3 - 2| - 8 = -7\), so you can plot point (3, -7).

Step 5: Draw the Graph

With the plot points , especially the vertex and a few additional points, we have enough to draw the 'V' shape produced by absolute value functions. The graph will bottom out at the vertex and then rise equally on either side of it.

Key Concepts

Vertex of Absolute Value Function

The vertex of an absolute value function is a fundamental concept in graphing. It is the 'turning point' of the graph where the direction changes, forming a distinct 'V' shape. For the standard function \(f(x) = |x|\), the vertex is at the origin \((0,0)\).

However, when the function is altered to \(f(x) = |x - h| + k\), the vertex shifts to the point \((h, k)\). In our exercise, the function is \(f(x) = |x - 2| - 8\). Following our transformation rules, we identify the vertex of this function to be \((2, -8)\). The '2' indicates a horizontal shift to the right, and the '-8' a vertical shift downwards. Understanding the vertex's location is crucial as it serves as a starting point for plotting the graph of the absolute value function.

Transformations of Functions

Transformations alter the appearance of the original function's graph without changing its overall shape. These manipulations include translations, reflections, stretches, and compressions.

For absolute value functions, transformations are relatively straightforward to visualize:

• A horizontal shift occurs if you add or subtract a constant from the \(x\) variable inside the absolute value, as \(f(x) = |x - h|\) moves the graph \(h\) units horizontally.
• A vertical shift results from adding or subtracting a constant outside the absolute value, represented as \(f(x) = |x| + k\), and this moves the graph \(k\) units vertically.

If we apply other transformations such as reflections over the x-axis \(f(x) = -|x|\) or y-axis inversions, the graph will respectively flip upside down or mirror along the y-axis. Recognizing these transformations helps in piecing together the final graph from the base function.

Plotting Absolute Value Function

After identifying the vertex and transformations, plotting an absolute value function becomes much simpler. Begin with the vertex – the most crucial point – then choose additional points to establish the 'V' shape symmetry.

For \(f(x) = |x-2|-8\), we plot the vertex \((2, -8)\) as our hinge point. To find other points, pick values of \(x\) around the vertex and calculate their \(f(x)\) values. For instance, at \(x = 1\) and \(x = 3\), we get the same \(f(x)\) value of -7 due to the symmetrical nature of the graph around the vertex. These points further confirm the 'V' shape and guide us on how the graph will look. Remember, each side of the vertex should be a mirror image of the other, creating a clear, sharp 'V' that characterizes the absolute value graph. With the vertex and a few plotted points, the absolute value graph can be sketched accurately with its linear arms extending infinitely outward from the vertex in both directions.