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Find the coordinates of the vertex for the parabola defined by the given quadratic function. f(x)=2x2−8x+3
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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 12
Identify which graphs are not those of polynomial functions.
Guida verificata passo dopo passo1
Step 1: Observe the shape of the graph. The graph shows a V-shaped function with two straight lines meeting at the origin (0,0).
Step 2: Recall that polynomial functions are smooth and continuous with no sharp corners or cusps. Their graphs are made up of curves, not sharp angles.
Step 3: Notice that the graph has a sharp corner at the origin, which indicates a cusp or a point where the function is not differentiable.
Step 4: Recognize that this V-shaped graph resembles the absolute value function, which is defined as \(y = |x|\), and is not a polynomial because it involves an absolute value operation.
Step 5: Conclude that since the graph has a sharp corner and is not smooth, it is not the graph of a polynomial function.
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Polynomial Functions
Polynomial functions are algebraic expressions consisting of variables raised to non-negative integer powers with constant coefficients. Their graphs are smooth and continuous curves without sharp corners or cusps, and they can have turning points but no abrupt changes in direction.
Video consigliato:
06:04
Introduction to Polynomial Functions
Graph Characteristics of Polynomial Functions
The graphs of polynomial functions are continuous and differentiable everywhere, meaning they have no sharp points or corners. They typically exhibit smooth curves, and their end behavior depends on the leading term's degree and coefficient.
Video consigliato:
05:25Graphing Polynomial Functions
Non-Polynomial Graphs and Sharp Corners
Graphs with sharp corners or 'V' shapes, like the absolute value function, are not polynomial functions because polynomials are differentiable everywhere. Such graphs indicate piecewise or non-polynomial functions, which have points where the derivative does not exist.
Video consigliato:
05:25Graphing Polynomial Functions
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