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 51
Capitolo 4, Problema 51

Write an equation in vertex form of the parabola that has the same shape as the graph of f(x) = 2x2 but with the given point as the vertex. (−10, −5)

Guida verificata passo dopo passo
1
Recall that the vertex form of a parabola is given by the equation \(y = a(x - h)^2 + k\), where \((h, k)\) is the vertex of the parabola and \(a\) determines the shape (width and direction) of the parabola.
Identify the value of \(a\) from the original function \(f(x) = 2x^2\). Here, \(a = 2\), which means the parabola opens upward and is narrower than the standard parabola \(y = x^2\).
Use the given vertex point \((-10, -5)\) to substitute \(h = -10\) and \(k = -5\) into the vertex form equation, so it becomes \(y = 2(x - (-10))^2 + (-5)\).
Simplify the expression inside the parentheses: \(x - (-10)\) becomes \(x + 10\), so the equation is \(y = 2(x + 10)^2 - 5\).
This equation represents a parabola with the same shape as \(f(x) = 2x^2\) but shifted so that its vertex is at \((-10, -5)\).

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:
1m

Concetti chiave

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

Vertex Form of a Quadratic Function

The vertex form of a quadratic function is expressed as f(x) = a(x - h)^2 + k, where (h, k) is the vertex of the parabola. This form makes it easy to identify the vertex and understand the parabola's position and shape.
Video consigliato:
08:07
Vertex Form

Effect of the 'a' Coefficient on Parabola Shape

The coefficient 'a' in a quadratic function affects the parabola's width and direction. If |a| > 1, the parabola is narrower; if 0 < |a| < 1, it is wider. A positive 'a' opens upward, while a negative 'a' opens downward.
Video consigliato:
5:28
Horizontal Parabolas

Using a Given Vertex to Write the Equation

When given a vertex and a shape defined by 'a', substitute the vertex coordinates (h, k) into the vertex form and use the given 'a' value to write the equation. This ensures the parabola has the correct shape and vertex location.
Video consigliato:
04:27
Finding Equations of Lines Given Two Points
Pratica correlata
Domanda del libro di testo

Find all zeros of the polynomial function or solve the given polynomial equation. Use the Rational Zero Theorem, Descartes's Rule of Signs, and possibly the graph of the polynomial function shown by a graphing utility as an aid in obtaining the first zero or the first root. 2x5+7x4−18x2−8x+8=0

603
views
Domanda del libro di testo
In Exercises 49–50, find all the zeros of each polynomial function and write the polynomial as a product of linear factors. g(x) = x^4 - 6x^3 + x^2 + 24x + 16
732
views
Domanda del libro di testo

In Exercises 51–54, graphs of fifth-degree polynomial functions are shown. In each case, specify the number of real zeros and the number of imaginary zeros. Indicate whether there are any real zeros with multiplicity other than 1.

704
views
Domanda del libro di testo

Use transformations of f(x)=1/x or f(x)=1/x2 to graph each rational function. g(x)=1/(x+2)2

945
views
Domanda del libro di testo

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. (x+4)/x>0

531
views
Domanda del libro di testo

Solve each rational inequality in Exercises 43–60 and graph the solution set on a real number line. Express each solution set in interval notation. x/(x−3)>0

557
views