Skip to main content
Ch. 3 - Polynomial and Rational Functions
Blitzer - College Algebra 8th Edition
Blitzer8th EditionCollege AlgebraISBN: 9780136970514Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoBlitzer 8th EditionCh. 3 - Polynomial and Rational FunctionsProblema 33
Capitolo 4, Problema 33

In Exercises 17–38, 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)=x2+6x+3

Guida verificata passo dopo passo
1
Identify the quadratic function: \(f(x) = x^{2} + 6x + 3\).
Find the vertex using the formula for the x-coordinate of the vertex: \(x = -\frac{b}{2a}\). Here, \(a = 1\) and \(b = 6\), so calculate \(x = -\frac{6}{2 \times 1}\).
Substitute the x-coordinate of the vertex back into the function to find the y-coordinate: \(f\left(-\frac{6}{2}\right) = \left(-\frac{6}{2}\right)^{2} + 6 \times \left(-\frac{6}{2}\right) + 3\).
Find the y-intercept by evaluating \(f(0)\), which is simply the constant term in this case: \(f(0) = 3\).
Find the x-intercepts by solving the quadratic equation \(x^{2} + 6x + 3 = 0\) using the quadratic formula: \(x = \frac{-b \pm \sqrt{b^{2} - 4ac}}{2a}\), where \(a=1\), \(b=6\), and \(c=3\).

Risposta video verificata per un problema simile:

Questa soluzione video è stata consigliata dai nostri tutor come utile per risolvere questo problema.
Durata del video:
2m

Concetti chiave

Ecco i concetti essenziali che devi comprendere per rispondere correttamente alla domanda.

Vertex of a Quadratic Function

The vertex is the highest or lowest point on the parabola, representing its maximum or minimum value. It can be found using the formula x = -b/(2a) for a quadratic function f(x) = ax² + bx + c. The vertex coordinates help in sketching the graph and understanding the parabola's shape.
Video consigliato:
08:07
Vertex Form

Axis of Symmetry

The axis of symmetry is a vertical line that passes through the vertex, dividing the parabola into two mirror-image halves. Its equation is x = -b/(2a). This line helps in graphing the parabola and identifying symmetric points on either side.
Video consigliato:
07:42
Properties of Parabolas

Domain and Range of a Quadratic Function

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.
Video consigliato:
4:22
Domain & Range of Transformed Functions
Pratica correlata
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. (2−x)2(x−7/2)<0

757
views
Domanda del libro di testo

Use synthetic division and the Remainder Theorem to find the indicated function value. f(x)=2x3−11x2+7x−5;f(4)

514
views
Domanda del libro di testo

Find the vertical asymptotes, if any, and the values of x corresponding to holes, if any, of the graph of each rational function. h(x)=(x+7)/(x2+4x−21)

1051
views
Domanda del libro di testo
Given f(x) = 2x^3 - 7x^2 + 9x - 3, use the Remainder Theorem to find f(- 13).
1134
views
Domanda del libro di testo

Solve each polynomial inequality in Exercises 1–42 and graph the solution set on a real number line. Express each solution set in interval notation. x(4−x)(x−6)≤0

496
views
Domanda del libro di testo

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)=2x−x2+3

953
views