Ch. 3 - Polynomial and Rational Functions

Lial13th EditionCollege AlgebraISBN: 9780136881063Not the one you use?Change textbook

Lial 13th Edition

Ch. 3 - Polynomial and Rational Functions

Problem 84

Chapter 4, Problem 84

Define the quadratic function ƒ having x-intercepts (1, 0) and (-2, 0) and y-intercept (0, 4).

Verified step by step guidance

Recall that a quadratic function with x-intercepts at points $(r_1, 0)$ and $(r_2, 0)$ can be expressed in factored form as: \[f(x) = a(x - r_1)(x - r_2)\] where $a$ is a constant that affects the vertical stretch or compression of the parabola.

Substitute the given x-intercepts $r_1 = 1$ and $r_2 = -2$ into the factored form: \[f(x) = a(x - 1)(x + 2)\]

Use the y-intercept $(0, 4)$ to find the value of $a$. Substitute $x = 0$ and $f(0) = 4$ into the equation: \[4 = a(0 - 1)(0 + 2)\]

Simplify the right side of the equation to solve for $a$: \[4 = a(-1)(2) = -2a\] Then solve for $a$ by dividing both sides by $-2$: \[a = \frac{4}{-2}\]

Write the final quadratic function by substituting the value of $a$ back into the factored form: \[f(x) = a(x - 1)(x + 2)\]

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Key Concepts

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

Quadratic Function Standard Form

A quadratic function is typically written as f(x) = ax^2 + bx + c, where a, b, and c are constants and a ≠ 0. This form helps identify the shape and position of the parabola on the coordinate plane.

Using x-Intercepts to Find Factors

The x-intercepts of a quadratic function correspond to the roots of the equation f(x) = 0. If the roots are r1 and r2, the function can be expressed as f(x) = a(x - r1)(x - r2), which helps in constructing the quadratic from given zeros.

Determining the Leading Coefficient Using the y-Intercept

The y-intercept is the value of f(x) when x = 0, which equals c in the standard form. By substituting x = 0 and the given y-intercept value into the factored form, we can solve for the leading coefficient a to fully define the quadratic function.

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Textbook Question

Solve each inequality. Give the solution set in interval notation. (x+7)/(2x+1)<0

Textbook Question

Graph each rational function. ƒ(x)=(x²+8x+16)/(x²+4x-5)

Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $ƒ (x) = 5 x^{4} + 3 x^{2} + 2 x - 9$

Textbook Question

Determine the different possibilities for the numbers of positive, negative, and nonreal complex zeros of each function. $ƒ (x) = 3 x^{4} + 2 x^{3} - 8 x^{2} - 10 x - 1$

Textbook Question

The distance between the two points $P of open paren x sub 1 comma y sub 1 close paren$ and $R of open paren x sub 2 comma y sub 2 close paren$ is $d(P, R) = $\sqrt{(x_1 - x_2)^2 + (y_1 - y_2)^2}$$ . Distance formula. Find the closest point on the line $y equals 2 x$ to the point $open paren 1 comma 7 close paren$ . (Hint: Every point on $y equals 2 x$ has the form $open paren x comma 2 x close paren$ , and the closest point has the minimum distance.)

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Textbook Question

Graph each rational function. ƒ(x)=(x²+2x+1)/(x²-x-6)