The graph of $h(x) = -\log_5(x+5)$. Find the coordinates of the intercepts and the equation of the asymptote of $h(x)$. Explain how to find these using the graph.
Answer (2)
Find the x-intercept by setting h ( x ) = 0 and solving for x , which gives x = − 4 .
Find the y-intercept by setting x = 0 and solving for h ( 0 ) , which gives h ( 0 ) = − 1 .
Find the vertical asymptote by setting the argument of the logarithm to zero, x + 5 = 0 , which gives x = − 5 .
The x-intercept is ( − 4 , 0 ) , the y-intercept is ( 0 , − 1 ) , and the vertical asymptote is x = − 5 . x-intercept: ( − 4 , 0 ) , y-intercept: ( 0 , − 1 ) , asymptote: x = − 5
Explanation
Understanding the Problem We are given the function h ( x ) = − lo g 5 ( x + 5 ) and asked to find the intercepts and asymptote.
Finding the x-intercept To find the x-intercept, we set h ( x ) = 0 and solve for x . This means we solve the equation − lo g 5 ( x + 5 ) = 0 . Multiplying both sides by − 1 , we have lo g 5 ( x + 5 ) = 0 . Since any number raised to the power of 0 is 1, we can rewrite this as x + 5 = 5 0 = 1 . Solving for x , we get x = 1 − 5 = − 4 . So the x-intercept is ( − 4 , 0 ) .
Finding the y-intercept To find the y-intercept, we set x = 0 and solve for h ( 0 ) . We have h ( 0 ) = − lo g 5 ( 0 + 5 ) = − lo g 5 ( 5 ) = − 1 . So the y-intercept is ( 0 , − 1 ) .
Finding the Vertical Asymptote To find the vertical asymptote, we need to find the value of x for which the argument of the logarithm is zero. That is, we need to solve x + 5 = 0 , which gives x = − 5 . The vertical asymptote is x = − 5 .
Graphical Interpretation On the graph, the x-intercept is the point where the graph intersects the x-axis, i.e., where h ( x ) = 0 . The y-intercept is the point where the graph intersects the y-axis, i.e., where x = 0 . The vertical asymptote is the vertical line that the graph approaches but never touches. It occurs where the argument of the logarithm approaches zero.
Final Answer Therefore, the x-intercept is ( − 4 , 0 ) , the y-intercept is ( 0 , − 1 ) , and the vertical asymptote is x = − 5 .
Examples
Logarithmic functions are used in many real-world applications, such as measuring the intensity of earthquakes on the Richter scale, modeling population growth, and determining the pH of a solution in chemistry. Understanding intercepts and asymptotes helps us analyze and interpret these models effectively. For example, in earthquake intensity measurement, the x-intercept could represent the minimum intensity detectable by a particular instrument, while the asymptote could indicate a theoretical upper limit of intensity based on the model's assumptions. Let's say the intensity of an earthquake is given by I ( t ) = − l o g 2 ( t + 16 ) , where t is the time in seconds. The asymptote would be t = − 16 , and the x-intercept would be when I ( t ) = 0 , which means − l o g 2 ( t + 16 ) = 0 , so t + 16 = 1 , and t = − 15 .
The x-intercept of the function h ( x ) = − lo g 5 ( x + 5 ) is ( − 4 , 0 ) , the y-intercept is ( 0 , − 1 ) , and the vertical asymptote is at x = − 5 .
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