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Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=−2(x+1)2+5
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Ch. 3 - Polynomial and Rational Functions
Blitzer8th EditionCollege AlgebraISBN: 9780136970514
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TOC.tableOfContents
- Ch. P - Fundamental Concepts of Algebra311
- Ch. 1 - Equations and Inequalities398
- Ch. 2 - Functions and Graphs407
- Ch. 3 - Polynomial and Rational Functions306
- Ch. 4 - Exponential and Logarithmic Functions247
- Ch. 5 - Systems of Equations and Inequalities173
- Ch. 6 - Matrices and Determinants143
- Ch. 7 - Conic Sections124
- Ch. 8 - Sequences, Induction, and Probability251
Capitolo 4, Problema 11
Use the graph of the rational function in the figure shown to complete each statement in Exercises 9–14.
As ______
Guida verificata passo dopo passo1
Identify the point of interest on the x-axis, which is \(x \to 1^-\), meaning we are approaching 1 from the left side.
Look at the graph near \(x = 1\) and observe the behavior of the function \(f(x)\) as \(x\) approaches 1 from values less than 1.
Notice that the graph is very close to the horizontal asymptote \(y = 0\) near \(x = 1\), and the function values are slightly below zero.
Since the function values are approaching zero from the negative side as \(x\) approaches 1 from the left, we conclude that \(f(x) \to 0^-\) as \(x \to 1^-\).
Therefore, the limit of \(f(x)\) as \(x\) approaches 1 from the left is 0, but the function values are slightly negative.
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Vertical Asymptotes
Vertical asymptotes occur where a function approaches infinity or negative infinity as the input approaches a specific value. They indicate values of x where the function is undefined, often due to division by zero in rational functions. In the graph, vertical asymptotes are shown at x = 6 and x = 14.
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3:12
Determining Vertical Asymptotes
Horizontal Asymptotes
A horizontal asymptote represents the value that a function approaches as x approaches positive or negative infinity. It shows the end behavior of the function. In this graph, the horizontal asymptote is y = 0, meaning the function values get closer to zero as x becomes very large or very small.
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4:48Determining Horizontal Asymptotes
Limit Behavior Near a Point
The limit of a function as x approaches a specific value from the left or right describes the function's behavior near that point. For example, as x approaches 1 from the left (x → 1⁻), the function value approaches a certain number or infinity. Understanding this helps in analyzing function continuity and asymptotic behavior.
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05:01Identifying Intervals of Unknown Behavior
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