Problem 29
Question
For the following exercises, graph the given functions by hand. \(f(x)=-|x-1|-3\)
Step-by-Step Solution
Verified Answer
Graph the inverted 'V' with vertex at (1, -3), created by shifting right 1 and down 3.
1Step 1: Understand the basic shape
The function given is of the form \[ f(x) = -|x - 1| - 3 \]which resembles the absolute value function. Normally, the graph of \( y = |x| \) is a 'V' shape. The function \( f(x) = -|x - 1| \) inverts the 'V' to an 'A' shape since the absolute value is negated.
2Step 2: Discover the transformation
The function \( f(x) = -|x - 1| - 3 \) involves a horizontal shift and a vertical shift. The term \( x - 1 \) indicates a shift to the right by 1 unit, and the \( -3 \) at the end indicates a vertical shift down by 3 units. This modifies the base function.
3Step 3: Identify critical points
Determine the vertex of the 'A' shape, which is the critical point where the absolute value changes direction. For this function, that point occurs when \( x = 1 \), giving the vertex at \( (1, -3) \). This is because substituting \( x = 1 \) into \( f(x) = -|x - 1| - 3 \) yields \( f(1) = -|0| - 3 = -3 \).
4Step 4: Plot the vertex
On a set of axes, plot the vertex point \( (1, -3) \). This is the lowest point on the 'A' shaped graph since the 'V' is flipped upside down.
5Step 5: Draw the 'A' shape
From the vertex, draw lines with a slope of \(-1\) to the left and right, creating an 'A' shaped graph. The line moving left from the vertex has the equation \( -x + 1 - 3 \) for values \( x < 1 \), and the line moving right has the equation \( -x + 1 - 3 \) for values \( x > 1 \).
6Step 6: Finalize the graph
Ensure the graph accurately represents the function and its transformations. The critical plot points should demonstrate behavior above the x-axis before crossing down at the vertex (1, -3) and forming the inverted 'V' shape.
Key Concepts
Vertex TransformationHorizontal ShiftVertical ShiftInverting Absolute Value Graphs
Vertex Transformation
The vertex of an absolute value function is a pivotal point where the graph changes its direction. In terms of a basic absolute value function like \( y = |x| \), this vertex is located at the origin, \( (0,0) \). However, when transformations are applied, the vertex's position alters accordingly. For instance, in the function \( f(x) = -|x-1|-3 \), the vertex is shifted rightwards and downwards.
To determine the vertex, we focus on the expression inside the absolute value, \( x - 1 \). Setting it to zero gives us the x-coordinate of the vertex, thus \( x = 1 \). The y-coordinate comes from evaluating the function at this point, resulting in \( f(1) = -3 \).
Therefore, the transformed vertex is at \( (1, -3) \). This point is crucial as it marks the lowest tip of the graph, acting as a cornerstone for sketching the rest of the graph.
To determine the vertex, we focus on the expression inside the absolute value, \( x - 1 \). Setting it to zero gives us the x-coordinate of the vertex, thus \( x = 1 \). The y-coordinate comes from evaluating the function at this point, resulting in \( f(1) = -3 \).
Therefore, the transformed vertex is at \( (1, -3) \). This point is crucial as it marks the lowest tip of the graph, acting as a cornerstone for sketching the rest of the graph.
Horizontal Shift
Horizontal shifts in graphs are part of the transformations affecting their positioning along the x-axis. These shifts occur when a constant is subtracted or added inside the absolute value function's formula. For the given function, \( f(x) = -|x - 1| - 3 \), the horizontal shift is found in the \( x - 1 \) part.
Specifically, \( x - 1 \) indicates a rightward shift by one unit. To see why consider what happens if you set the expression inside the absolute value to zero. Solving \( x - 1 = 0 \) gives \( x = 1 \). This tells us that the whole graph moves to the right from its usual position.
This transformation does not alter the graph's overall shape but only its left-right location on the plane. As a student visualizing this graph, always think about how the function's rules cause these shifts without stretching or compressing the visual form of the graph.
Specifically, \( x - 1 \) indicates a rightward shift by one unit. To see why consider what happens if you set the expression inside the absolute value to zero. Solving \( x - 1 = 0 \) gives \( x = 1 \). This tells us that the whole graph moves to the right from its usual position.
This transformation does not alter the graph's overall shape but only its left-right location on the plane. As a student visualizing this graph, always think about how the function's rules cause these shifts without stretching or compressing the visual form of the graph.
Vertical Shift
Vertical shifts happen when a number is added or subtracted from the whole function. In the equation \( f(x) = -|x - 1| - 3 \), this shift is represented by the \(-3\) at the end.
In simple terms, vertical shifts move every point of the graph up or down the y-axis, maintaining the original shape like a shadow cast over a wall. Here, the \(-3\) means the entire graph shifts three units downward.
This downward movement affects all points, shifting every possible output of the original absolute value function. Therefore, where once, with a basic absolute value function, the vertex might have sat at the origin, in this instance, it has dipped down to \( (1, -3) \), as noted from our vertex calculations.
It's helpful to visualize this as lowering the y-values of the vertex and reflecting the function across its usual horizontal symmetry.
In simple terms, vertical shifts move every point of the graph up or down the y-axis, maintaining the original shape like a shadow cast over a wall. Here, the \(-3\) means the entire graph shifts three units downward.
This downward movement affects all points, shifting every possible output of the original absolute value function. Therefore, where once, with a basic absolute value function, the vertex might have sat at the origin, in this instance, it has dipped down to \( (1, -3) \), as noted from our vertex calculations.
It's helpful to visualize this as lowering the y-values of the vertex and reflecting the function across its usual horizontal symmetry.
Inverting Absolute Value Graphs
When dealing with an absolute value function, inversion transforms the shape from a 'V' to an 'A'. This happens by negating the function outside the absolute value. Consider the function \( f(x) = -|x - 1| - 3 \). The key inversion factor here is the minus sign before the absolute value.
Such a change means that all outputs which would have been positive (above the x-axis) become negative (below the x-axis), creating an upside-down version of the typical 'V' shape.
An easy approach to identifying this inversion lies in noticing that wherever the graph would traditionally open upward, it now opens downward, creating a peak at its vertex.
This inverted 'V' or 'A' symbolizes how transformations alter the original path of the graph, making it essential to regard the minus sign as a signal for inverting the output in absolute value equations.
Such a change means that all outputs which would have been positive (above the x-axis) become negative (below the x-axis), creating an upside-down version of the typical 'V' shape.
An easy approach to identifying this inversion lies in noticing that wherever the graph would traditionally open upward, it now opens downward, creating a peak at its vertex.
This inverted 'V' or 'A' symbolizes how transformations alter the original path of the graph, making it essential to regard the minus sign as a signal for inverting the output in absolute value equations.
Other exercises in this chapter
Problem 28
For the following exercises, find the average rate of change of each function on the interval specified. \(f(x)=x^{2}\) on [1,5]
View solution Problem 28
For the following exercises, evaluate the function \(f\) at the indicated values \(f(-3), f(2), f(-a),-f(a), f(a+h)\). \(f(x)=-5 x^{2}+2 x-1\)
View solution Problem 29
For the following exercises, sketch a graph of the function as a transformation of the graph of one of the toolkit functions. \(k(x)=(x-2)^{3}-1\)
View solution Problem 29
For the following exercises, find functions \(f(x)\) and \(g(x)\) so the given function can be expressed as \(h(x)=f(g(x))\). \(h(x)=\frac{4}{(x+2)^{2}}\)
View solution