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Use the Rational Zero Theorem to list all possible rational zeros for each given function.
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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 35
Use the vertex and intercepts to sketch the graph of each quadratic function. Give the equation of the parabola's axis of symmetry. Use the graph to determine the function's domain and range. f(x)=2x2+4x−3
Guida verificata passo dopo passo1
Identify the quadratic function given: \(f(x) = 2x^{2} + 4x - 3\).
Find the vertex of the parabola using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\), where \(a = 2\) and \(b = 4\).
Calculate the y-coordinate of the vertex by substituting the x-value found into the function: \(f(x) = 2x^{2} + 4x - 3\).
Determine the axis of symmetry, which is the vertical line passing through the vertex, given by the equation \(x = -\frac{b}{2a}\).
Find the x-intercepts by solving the quadratic equation \(2x^{2} + 4x - 3 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), and find the y-intercept by evaluating \(f(0)\).
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Vertex of a Quadratic Function
The vertex of a quadratic function is the highest or lowest point on its graph, a parabola. It can be found using the formula x = -b/(2a) for a function in standard form f(x) = ax² + bx + c. The vertex helps determine the parabola's maximum or minimum value and is essential for sketching the graph accurately.
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08:07
Vertex Form
Axis of Symmetry
The axis of symmetry is a vertical line that passes through the vertex of the parabola, dividing it into two mirror-image halves. Its equation is x = -b/(2a). Knowing the axis of symmetry helps in graphing the parabola and understanding the symmetry of the function's values.
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07:42Properties of Parabolas
Domain and Range of Quadratic Functions
The domain of any quadratic function is all real numbers since x can take any value. The range depends on the vertex: if the parabola opens upward (a > 0), the range is all values greater than or equal to the vertex's y-coordinate; if it opens downward (a < 0), the range is all values less than or equal to the vertex's y-coordinate.
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4:22Domain & Range of Transformed Functions
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