Ch. 3 - Polynomial and Rational Functions

Blitzer8th EditionCollege AlgebraISBN: 9780136970514Not the one you use?Change textbook

Blitzer 8th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 97

Chapter 4, Problem 97

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-4) The graph passes through the point (1,4).

Verified step by step guidance

Recall that the vertex form of a parabola's equation is given by

y = a (x - h) 2 + k

, where (h, k) is the vertex of the parabola.

Substitute the vertex coordinates (-3, -4) into the vertex form equation, replacing h with -3 and k with -4, so the equation becomes

y = a (x + 3) 2 - 4

Use the point (1, 4) that lies on the parabola to find the value of a. Substitute x = 1 and y = 4 into the equation:

4 = a (1 + 3) 2 - 4

Simplify the expression inside the parentheses and the square:

4 = a (4) 2 - 4

becomes

4 = a imes 16 - 4

Solve the resulting equation for a by isolating a on one side: add 4 to both sides and then divide by 16 to find the value of a.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Vertex Form of a Parabola

The vertex form of a parabola's equation is y = a(x - h)^2 + k, where (h, k) is the vertex. This form makes it easy to identify the vertex and understand the parabola's shape and position on the coordinate plane.

Using a Point to Find the Parameter 'a'

After substituting the vertex coordinates into the vertex form, use another point on the parabola to solve for 'a'. This parameter controls the parabola's width and direction (upward if positive, downward if negative).

Substitution and Solving Quadratic Equations

Substitute the known point's x and y values into the vertex form equation to create an equation in terms of 'a'. Then solve this equation algebraically to find the exact value of 'a', completing the parabola's equation.

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Textbook Question

Use long division to rewrite the equation for g in the form quotient + remainder/divisor. Then use this form of the function's equation and transformations of f(x) = 1/x to graph g. g(x) = (2x+7)/(x+3)

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Textbook Question

In Exercises 97–98, write the equation of each parabola in vertex form. Vertex: (-3,-1) The graph passes through the point (-2,-3).

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Textbook Question

Find the average rate of change of f(x)=√x from x₁=4 to x₂=9.

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Textbook Question

In Exercises 89–94, the equation for f is given by the simplified expression that results after performing the indicated operation. Write the equation for f and then graph the function. (1 − 3/(x+2)) / (1 + 1/(x−2))

Solve each inequality in Exercises 86–91 using a graphing utility. 1/(x + 1) ≤ 2/(x + 4)

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