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Chapter 1: Problem 16

Find the ordered pairs that represent solutions to cach of the following equations when \(x=0\), when \(x=3\), and when \(x=-2\) a. \(y=2 x^{2}+5 x\) c. \(y=x^{3}+x^{2}\) b. \(y=-x^{2}+1\) d. \(y=3(x-2)(x-1)\)

Short Answer

Expert verified
Ordered pairs: (a) (0, 0), (3, 33), (-2, -2); (b) (0, 1), (3, -8), (-2, -3); (c) (0, 0), (3, 36), (-2, -4); (d) (0, 6), (3, 6), (-2, 36)

Step by step solution

01

Evaluate Equation a for Given x Values

Substitute the given values of x (0, 3, -2) into the equation a: \(y=2 x^{2}+5 x\).When \(x=0\): \(y = 2(0)^2 + 5(0) = 0\). The ordered pair is (0, 0).When \(x=3\): \(y = 2(3)^2 + 5(3) = 2(9) + 15 = 18 + 15 = 33\). The ordered pair is (3, 33).When \(x=-2\): \(y = 2(-2)^2 + 5(-2) = 2(4) + (-10) = 8 - 10 = -2\). The ordered pair is (-2, -2).
02

Evaluate Equation b for Given x Values

Substitute the given values of x (0, 3, -2) into the equation b: \(y=-x^{2}+1\).When \(x=0\): \(y = -(0)^2 + 1 = 1\). The ordered pair is (0, 1).When \(x=3\): \(y = -(3)^2 + 1 = -9 + 1 = -8\). The ordered pair is (3, -8).When \(x=-2\): \(y = -(-2)^2 + 1 = -(4) + 1 = -4 + 1 = -3\). The ordered pair is (-2, -3).
03

Evaluate Equation c for Given x Values

Substitute the given values of x (0, 3, -2) into the equation c: \(y=x^{3}+x^{2}\).When \(x=0\): \(y = (0)^3 + (0)^2 = 0\). The ordered pair is (0, 0).When \(x=3\): \(y = (3)^3 + (3)^2 = 27 + 9 = 36\). The ordered pair is (3, 36).When \(x=-2\): \(y = (-2)^3 + (-2)^2 = -8 + 4 = -4\). The ordered pair is (-2, -4).
04

Evaluate Equation d for Given x Values

Substitute the given values of x (0, 3, -2) into the equation d: \(y=3(x-2)(x-1)\).When \(x=0\): \(y = 3(0-2)(0-1) = 3(-2)(-1) = 6\). The ordered pair is (0, 6).When \(x=3\): \(y = 3(3-2)(3-1) = 3(1)(2) = 6\). The ordered pair is (3, 6).When \(x=-2\): \(y = 3(-2-2)(-2-1) = 3(-4)(-3) = 36\). The ordered pair is (-2, 36).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

ordered pairs
Ordered pairs are a fundamental concept in coordinate geometry. They represent a point in a two-dimensional space, consisting of an x-coordinate and a y-coordinate written as (x, y). For example, in the problem, we are given equations and asked to find the ordered pairs for specific values of x. The method involves substituting the given x values into the equation to find the corresponding y values. For instance, in equation a, when x=0, y=0, which gives us the ordered pair (0, 0). Understanding ordered pairs is crucial because they allow us to graph equations and visualize relationships between variables.
substitution method
The substitution method is a straightforward way to solve equations, especially useful when finding ordered pairs. This method involves replacing a variable in an equation with a given value. For instance, in the given problem, we substitute the values x=0, x=3, and x=-2 into each equation to find corresponding y values. Using equation b, when substituting x=3: y = -(3)^2 + 1, we find y = -9 + 1 = -8. Thus, the ordered pair is (3, -8). This method simplifies the problem by breaking it into manageable steps, making it easier to solve complex equations as we only deal with one variable at a time.
quadratic equations
Quadratic equations are a type of polynomial equation of degree 2 and have the general form: ax^2 + bx + c = 0. In the problem, equations a and b are quadratic equations: y=2x^2+5x and y=-x^2+1. Let's explore equation a. To find the ordered pair for x=3, we substitute and get y = 2(3)^2 + 5(3) = 18 + 15 = 33, resulting in the pair (3, 33). Quadratics often produce a parabolic graph, either opening upwards or downwards depending on the leading coefficient (a). Understanding how to handle quadratic equations is essential for solving a broad range of algebraic problems.
cubic equations
Cubic equations are polynomial equations of degree 3 and can be written in the general form: ax^3 + bx^2 + cx + d = 0. In this problem, equation c is a cubic equation: y=x^3+x^2. To find the ordered pair for x=-2, we substitute and get y = (-2)^3 + (-2)^2 = -8 + 4 = -4, resulting in the pair (-2, -4). Cubic equations can have one or more turning points, creating multiple curves in their graph. They are more complex than quadratic equations and learning to solve them prepares students for advanced mathematical topics.

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Most popular questions from this chapter

Write a formala to express each of the following sentences: a. The sale price is \(20 \%\) off the original price. Use \(S\) for sale price and \(P\) for original price to express \(S\) as a function of \(P\). b. The time in Paris is 6 hours ahead of New York. Use \(P\) for Paris time and \(N\) for New York time to express \(P\) as a function of \(N\). (Represent your answer in terms of a 12 -hour clock.) How would you adjust your formula if \(\bar{P}\) comes out preater than \(12 ?\) c. For temperatures above \(0^{\circ} \mathrm{F}\) the wind chill effect can be estimated by subtracting two-thirds of the wind speed (in miles per hour) from the outdoor temperature. Use \(C\) for the effective wind chill temperature, \(W\) for wind speed, and \(T\) for the actual outdoor temperature to write an equation expressing \(C\) in terms of \(W\) and \(T\).

(Use of calculator or other technology recommended.) Use the following table to generate an estimate of the mean age of the U.S. population. Show your work. (Hint: Replace each age interval with an age approximately in the middle of the interval.) $$ \begin{aligned} &\text { Ages }\\\ &\text { of US. Population in } 2004\\\ &\begin{array}{lr} \hline \begin{array}{c} \text { Age } \\ \text { (years) } \end{array} & \begin{array}{c} \text { Population } \\ \text { (thousands) } \end{array} \\ \hline \text { Under 10 } & 39,677 \\ 10-19 & 41,875 \\ 20-29 & 40,532 \\ 30-39 & 41,532 \\ 40-49 & 45,179 \\ 50-59 & 35,986 \\ 60-74 & 31,052 \\ 75-84 & 12,971 \\ 85 \text { and over } & 4,860 \\ \hline \text { Total } & 293,655 \\ \hline \end{array} \end{aligned} $$

Sketch a plausible graph for each of the following and label the axes. a. The amount of snow in your backyard each day from December 1 to March 1 b. The temperature during a 24 -hour period in your home town during one day in July. c. The amount of water inside your fishing boat if your boat leaks a little and your fishing partner bails out water every once in a while. d. The total hours of daylight each day of the year. e. The temperature of an ice-cold drink left to stand.

If \(f(x)=(2 x-1)^{2},\) evaluate \(f(0), f(1),\) and \(f(-2)\)

Given here is a table of salaries taken from a survey of recent graduates (with bachelor degrees) from a well-known university in Pittsburgh. $$ \begin{array}{cc} \hline \begin{array}{c} \text { Salary } \\ \text { (in thousands) } \end{array} & \begin{array}{c} \text { Number of Graduates } \\ \text { Receiving Salary } \end{array} \\ \hline 21-25 & 2 \\ 26-30 & 3 \\ 31-35 & 10 \\ 36-40 & 20 \\ 41-45 & 9 \\ 46-50 & 1 \\ \hline \end{array} $$ a. How many graduates were surveyed? \(\mathbf{b}\). Is this quantitative or qualitative data? Explain. c. What is the relative frequency of people having a salary between \(\$ 26,000\) and \(\$ 30,000 ?\) d. Create a histogram of the data.
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