Problem 58 Write an equation for a function... [FREE SOLUTION]

91影视

Precalculus: Graphs and Models, A Right Triangle Approach

Marvin L. Bittinger, Judith A. Beecher, David J. Ellenbogen

$Math Studyset 91影视 Explanations$ Math

6 Edition

Chapter 2: Problem 58

Write an equation for a function that has a graph with the given characteristics. The shape of $y=1 / x,$ but reflected across the $x$ -axis and shifted up 1 unit

Short Answer

Expert verified

The equation is $ y = -\frac{1}{x} + 1 $.

Step by step solution

Identify the base function

The base function given is the reciprocal function: \[ y = \frac{1}{x} \]

Reflect across the x-axis

To reflect the function across the x-axis, multiply the function by -1: \[ y = -\frac{1}{x} \]

Shift up 1 unit

To shift the function up by 1 unit, add 1 to the function: So the resulting function is: \[ y = -\frac{1}{x} + 1 \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Reflection of Functions

Reflecting a function across the x-axis changes the sign of its output values. This means you multiply the entire function by -1. Imagine flipping a graph upside down along the horizontal axis. For instance, if you have a base function like $y = \frac{1}{x}$, reflecting it across the x-axis would result in $y = -\frac{1}{x}$. This is because every positive value becomes negative, and every negative value becomes positive. Let's consider another example: the function $y = x^2$. Reflecting this function gives us $y = -x^2$. In general, if you have a function $f(x)$, its reflection across the x-axis is given by $y = -f(x)$.

Vertical Shift

Vertical shifts involve moving a graph up or down without changing its shape. You can visualize this as lifting a graph up or pressing it down. To shift a graph upwards, you add a constant to the function; to shift it downwards, you subtract a constant. For example:

If you start with $y = \frac{1}{x}$ and want to move it up by 1 unit, you would get $y = \frac{1}{x} + 1$.
Conversely, if you wanted to shift it down by 2 units, you would write $y = \frac{1}{x} - 2$.

So, in our exercise when we shift \-\frac{1}{x}\ up by 1 unit, we add 1 to it, resulting in $y = -\frac{1}{x} + 1$. You can apply this to any function $f(x)$: shifting it up by a units gives $y = f(x) + a$.

Reciprocal Function

A reciprocal function is a function of the form $y = \frac{1}{x}$. It has distinct traits: it is undefined at $x = 0$ because division by zero is impossible, and the graph approaches the x-axis and y-axis but never touches them. These approaches are called asymptotes.
When modifying functions, understanding the effects of transformation on the reciprocal function is essential. For instance:

Reflecting $y = \frac{1}{x}$ across the x-axis makes it $y = -\frac{1}{x}$, altering all the positive values to negative ones and vice versa.
Applying a vertical shift further modifies the function, such as adding 1 to give $y = -\frac{1}{x} + 1$.

Always remember the original form to trace these transformations easily.

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91影视

Write an equation for a function that has a graph with the given characteristics. The shape of \(y=1 / x,\) but reflected across the \(x\) -axis and shifted up 1 unit

Short Answer

Step by step solution

Identify the base function

Reflect across the x-axis

Shift up 1 unit

Key Concepts

Reflection of Functions

Vertical Shift

Reciprocal Function

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