Question

Determine whether each statement is always, sometimes, or never true. Explain your reasoning. a. A vertical translation of the graph of \(f(x)=\log x\) changes the equation of the asymptote. b. A vertical translation of the graph of \(f(x)=e^x\) changes the equation of the asymptote. c. A horizontal shrink of the graph of \(f(x)=\log x\) does not change the domain. d. The graph of \(g(x)=a b^{x-h}+k\) does not intersect the \(x\)-axis.

Short answer

a. Never true. b. Sometimes true. c. Always true. d. Sometimes true.

Step-by-step solution

Statement a

A vertical translation of the graph of \(f(x)=\log x\) implies a function of the form \(f(x)=\log x + c\), where c is the translation factor. No matter the value of c, the asymptote of the logarithmic function remains at \(x=0\). Therefore, a vertical translation of the graph of \(f(x)=\log x\) never changes the equation of the asymptote.

Statement b

A vertical translation of the graph of \(f(x)=e^x\) implies a function of the form \(f(x)=e^x + c\), where c is the translation factor. If c is not zero, the new function would have a horizontal asymptote other than \(y=0\). Therefore, a vertical translation of the graph of \(f(x)=e^x\) sometimes changes the equation of the asymptote.

Statement c

A horizontal shrink of the graph of \(f(x)=\log x\) implies a function of the form \(f(x)=\log(bx)\), where b is the shrink factor. This function will still be defined for \(x>0\), thus it does not change the domain of the function. Therefore, a horizontal shrink of the graph of \(f(x)=\log x\) always does not change the domain.

Statement d

The graph of \(g(x)=a b^{x-h}+k\) could intersect the x-axis depending on the values of a, b, h, and k. If \(a b^{-h}+k=0\), then the function intersects the x-axis. Therefore, the statement that the graph of \(g(x)=a b^{x-h}+k\) does not intersect the x-axis is sometimes true.

Key concepts

Vertical Translation

When we talk about vertical translation in mathematics, we're referring to a shift of a graph up or down along the y-axis. This shift doesn't affect certain fundamental properties, such as vertical asymptotes in some functions. For example, in the case of the logarithmic function \( f(x) = \log x \), translating the function vertically to \( f(x) = \log x + c \) keeps its vertical asymptote at \( x = 0 \).
However, if we take the exponential function \( f(x) = e^x \) and translate it vertically, the vertical shift does impact the asymptote. The original asymptote of \( y = 0 \) changes when the function becomes \( f(x) = e^x + c \), where the new asymptote moves to \( y = c \) when \( c eq 0 \).
Therefore, while vertical translations can sometimes affect horizontal asymptotes, they never affect vertical ones in the context of logarithmic functions.

Horizontal Shrink

Horizontal shrink involves compressing a graph towards the y-axis. This is achieved by multiplying the input \( x \) by a factor \( b > 1 \). When you apply a horizontal shrink to the logarithmic function \( f(x) = \log x \), it transforms to \( f(x) = \log(bx) \).
Despite this change, the domain remains unaffected. The original domain of the logarithm, \( x > 0 \), stays the same even after horizontal shrinkage. This is because the argument \( bx \) is still required to be greater than zero due to the nature of logarithms. Thus, a horizontal shrink won't affect the domain for the logarithmic function.

Logarithmic Function

A logarithmic function is a fundamental concept in mathematics, often written as \( f(x) = \log_b x \), where \( b \) is the base. The most common form is the natural logarithm, or \( \log x \), which assumes a base of \( e \), the natural number (~2.718).
These functions are unique because they have a vertical asymptote at \( x = 0 \), meaning the graph approaches this line but never touches or crosses it. They have domains of \( x > 0 \), reflecting that you can only take logarithms of positive numbers in real numbers.

• Vertical shifts do not alter the asymptote location.
• Horizontal transformations, like stretches or shrinks, maintain the domain.

Understanding these properties helps in manipulating and translating graphs correctly when solving mathematical problems involving logarithms.

Domain in Algebra

In algebra, the domain of a function represents all the input values (\( x \)-values) for which the function is defined. It's crucial for understanding how a function behaves and for solving equations and inequalities.
For instance, the function \( f(x) = \log x \) has a domain of \( x > 0 \). Any transformation applied to this function, such as \( f(x) = \log(bx) \) after a horizontal shrink, keeps this domain intact.

• Logarithmic functions are typically defined only for positive inputs.
• Polynomials, by contrast, have domains of all real numbers since any real number can be substituted into the polynomial's equation.
Being familiar with the domain helps distinguish which values will or won't work in specific functions, a key step in effective problem-solving.