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Chapter 9: Problem 4

Write an equation of a quadratic function that corresponds to each pair of \(x\)-intercepts. Assume there is no vertical stretch or shrink. a. \(2.5\) and \(-1\) (a) b. \(-4\) and \(-4\) c. \(-2\) and 2 d. \(r_{1}\) and \(r_{2}\)

Short Answer

Expert verified
a) \(x^2 - 1.5x - 2.5\); b) \(x^2 + 8x + 16\); c) \(x^2 - 4\); d) \(x^2 - (r_1+r_2)x + r_1r_2\).

Step by step solution

01

Understanding the standard form of a quadratic equation

A quadratic function can be expressed as \( f(x) = a(x-r_1)(x-r_2) \), where \( r_1 \) and \( r_2 \) are the \( x \)-intercepts. Since there is no vertical stretch or shrink, the standard form assumes \( a = 1 \).
02

Quadratic equation for intercepts 2.5 and -1

For \( x \)-intercepts \( 2.5 \) and \( -1 \), the quadratic equation is \( f(x) = (x-2.5)(x+1) \). Simplifying, we expand this to get \( f(x) = x^2 - 1.5x - 2.5 \).
03

Quadratic equation for intercepts -4 and -4

When both \( x \)-intercepts are the same, as with \( -4 \), the equation is \( f(x) = (x+4)^2 \). Expanding this gives \( f(x) = x^2 + 8x + 16 \).
04

Quadratic equation for intercepts -2 and 2

For \( x \)-intercepts \( -2 \) and \( 2 \), the equation is \( f(x) = (x+2)(x-2) \). This is the difference of squares, so \( f(x) = x^2 - 4 \).
05

Quadratic equation for intercepts \( r_1 \) and \( r_2 \)

For general \( x \)-intercepts \( r_1 \) and \( r_2 \), the quadratic equation is \( f(x) = (x-r_1)(x-r_2) \). Expanding this equation gives \( f(x) = x^2 - (r_1 + r_2)x + r_1r_2 \).

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Key Concepts

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

x-intercepts
The term **x-intercepts** is fundamental in understanding the behavior of quadratic functions. An x-intercept is a point where a graph intersects the x-axis. At these points, the value of the function, denoted as \(f(x)\), is zero. To find the x-intercepts of a quadratic function, we look at the equation in the form of \((x - r_1)(x - r_2)\), where \(r_1\) and \(r_2\) represent the x-intercepts.
  • Each intercept corresponds to a root of the quadratic equation.
  • For real and distinct intercepts, the function graph crosses the x-axis at two distinct points.
  • If the intercepts are identical, the graph will "touch" the x-axis at this point, representing a repeated root.
  • If there are no real intercepts, the parabola does not intersect the x-axis and the roots are complex numbers.
By recognizing the intercepts, you can quickly sketch the shape and position of a parabola along the x-axis. This insight is crucial for solving equations graphically and understanding quadratic behavior.
standard form of quadratic equation
The **standard form of a quadratic equation** is important for analyzing the structure of quadratic functions. Generally, it is represented as \(ax^2 + bx + c\). However, when finding x-intercepts and when there is no vertical stretch or shrink, it is more useful to express it as \(f(x) = a(x - r_1)(x - r_2)\). Here are key aspects:
  • The coefficient \(a\) determines the vertical stretch and the direction of the parabola. In this case, \(a = 1\).
  • The terms \(r_1\) and \(r_2\) in \((x - r_1)(x - r_2)\) denote where the graph intersects the x-axis.
  • Expanding \((x - r_1)(x - r_2)\) transforms it back into the standard form \(ax^2 + bx + c\).
To facilitate computations and graphing tasks, transitioning between the factored form and the standard form is a critical skill. This allows for an easy determination of intercepts and other properties of the function.
difference of squares
The **difference of squares** is a special method used in algebra to simplify expressions that can be factored into the form \((x + a)(x - a)\). This technique is particularly handy when dealing with quadratic expressions like \((x - 2)(x + 2)\), resulting in \(x^2 - 4\). This factoring is possible whenever you see the structure of \(a^2 - b^2\):
  • The formula is \(a^2 - b^2 = (a + b)(a - b)\).
  • This technique simplifies expressions by reducing them to the difference between two squares.
  • It is useful for solving equations rapidly, as it identifies the roots of an equation swiftly.
Utilizing this concept, equations such as \(f(x) = (x+2)(x-2)\), can be quickly rewritten and understood in the form \(x^2 - 4\), leading to faster solutions.
expanding polynomials
When working with quadratic functions, **expanding polynomials** is a technique used to simplify expressions, transforming them into a standard quadratic form. Given an expression like \((x - r_1)(x - r_2)\), expanding involves distributing each term to simplify the equation into a trinomial of the form \(ax^2 + bx + c\).
  • Start by applying the distributive property: \((a + b)(c + d) = ac + ad + bc + bd\).
  • Continue by collecting and combining like terms for simplicity.
  • By rewriting the equation, you ensure clarity and a consistent form for comparison and computation.
For example, \( (x - 2.5)(x + 1) \) expands to \( x^2 - 1.5x - 2.5 \), helping facilitate further analysis and applications. Expanding is a critical skill for algebraic manipulation and solving equations quickly and accurately.

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Most popular questions from this chapter

The equation \(h=-4.9 t^{2}+17 t+2.2\) models the height of a stone thrown into the air, where \(t\) is in seconds and \(h\) is in meters. Use the quadratic formula to find how long the stone is in the air.

Find the roots of each equation to the nearest thousandth by looking at a graph, zooming in on a table, or both. a. \(0=x^{2}+2 x-2\) (a) b. \(0=-3 x^{2}-4 x+3\)

Consider the equation \(y=x^{2}-9\). a. Graph the equation. What are the \(x\)-intercepts? b. Write the factored form of the equation. c. How are the \(x\)-intercepts related to the original equation? d. Write each equation in factored form. Verify each answer by graphing. i. \(y=x^{2}-49\) ii. \(y=16-x^{2}\) iii. \(y=x^{2}-47\) iv. \(y=x^{2}-28\) e. An expression in the form \(a^{2}-b^{2}\) is called a difference of two squares. Based on your work in \(12 \mathrm{a}-\mathrm{d}\), make a conjecture about the factored form of \(a^{2}-b^{2}\). f. Graph the equation \(y=x^{2}+4\). How many \(x\)-intercepts can you see? g. Explain the difficulty in trying to write the equation in \(12 f\) in factored form.

Determine whether each table represents a linear function, an exponential function, a cubic function, or a quadratic function. (Ti) $$ \begin{aligned} &\text { a. }\\\ &\begin{array}{|c|c|} \hline x & y \\ \hline 2 & 4 \\ \hline 5 & 25 \\ \hline 8 & 64 \\ \hline 11 & 121 \\ \hline 14 & 196 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { b. }\\\ &\begin{array}{|c|c|} \hline x & y \\ \hline 2 & 7 \\ \hline 5 & 11 \\ \hline 8 & 15 \\ \hline 11 & 19 \\ \hline 14 & 23 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { c. }\\\ &\begin{array}{|c|r|} \hline x & y \\ \hline 2 & 4 \\ \hline 5 & 32 \\ \hline 8 & 256 \\ \hline 11 & 2,048 \\ \hline 14 & 16,384 \\ \hline \end{array} \end{aligned} $$ $$ \begin{aligned} &\text { d. }\\\ &\begin{array}{|c|c|} \hline x & y \\ \hline 2 & 8 \\ \hline 5 & 125 \\ \hline 8 & 512 \\ \hline 11 & 1,331 \\ \hline 14 & 2,744 \\ \cline { 2 } & \\ \hline \end{array} \end{aligned} $$

The path of a ball in flight is given by \(p(x)=-0.23(x-3.4)^{2}+4.2\), where \(x\) is the horizontal distance in meters and \(p(x)\) is the vertical height in meters. Note that in this case the graph is the path of the ball, not the graph of the ball's height over time. a. Find \(p(2)\) and give a real-world meaning for this value. b. Find the \(x\)-values for \(p(x)=2\), and describe their real-world meanings. c. How high is the ball when it is released? d. How far will the ball travel horizontally before it hits the ground?
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