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Chapter 1: Problem 18

\(17 - 18\) Do the graphs intersect in the given viewing rectangle? If they do, how many points of intersection are there? $$y = 6 - 4 x - x ^ { 2 } , y = 3 x + 18 ; [ - 6,2 ]\( by \)[ - 5,20 ]$$

Short Answer

Expert verified
Yes, the graphs intersect at two points: \((-4, 6)\) and \((-3, 9)\).

Step by step solution

01

Set Equations Equal to Each Other

To find the points of intersection, set the two equations equal to each other: \(6 - 4x - x^2 = 3x + 18\). This will allow us to find the \(x\)-values at which the two graphs intersect.
02

Rearrange the Equation

Rearrange the equation from Step 1 to make it easier to solve: \(-x^2 - 7x - 12 = 0\). This is a standard form of a quadratic equation.
03

Solve the Quadratic Equation

To find the \(x\) values, we can use the quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\), where \( a = -1\), \( b = -7\), and \( c = -12\). Calculate the discriminant: \(b^2 - 4ac = (-7)^2 - 4(-1)(-12) = 49 - 48 = 1\). Since the discriminant is positive, there are two real solutions.
04

Calculate the Roots

Use the quadratic formula to find the roots: \(x = \frac{-(-7) \pm \sqrt{1}}{2(-1)}\). This simplifies to \(x = \frac{7 \pm 1}{-2}\). Solving this gives two solutions: \(x = -4\) and \(x = -3\).
05

Check if Roots Lie Within the Interval

The viewing rectangle only includes \(x\) values from \(-6\) to \(2\). Both \(x = -4\) and \(x = -3\) are within this interval, so both points are valid intersections within the given viewing rectangle.
06

Find Corresponding \(y\) Values

Substitute \(x = -4\) and \(x = -3\) back into either original equation to find the \(y\)-values. For \(y = 3x + 18\): when \(x = -4\), \(y = 3(-4) + 18 = 6\); when \(x = -3\), \(y = 3(-3) + 18 = 9\). Therefore, the points of intersection are \((-4, 6)\) and \((-3, 9)\).

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Key Concepts

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

Quadratic Equation Basics
A quadratic equation is an essential concept in algebra that takes the standard form \(ax^2 + bx + c = 0\). Here, \(a\), \(b\), and \(c\) are constants, and \(a eq 0\). The equation represents a parabola when graphed on a coordinate plane. Understanding the components of these equations helps in predicting the direction and the shape of the parabola.
  • The coefficient \(a\) influences the width and the direction of the parabola. A positive \(a\) makes it open upwards, while a negative \(a\) makes it open downwards.
  • The parameter \(b\) affects the parabola's direction but plays an obscure role in widening or narrowing.
  • The constant \(c\) determines the parabola's position on the y-axis.
Quadratic equations arise often, especially when finding intersections like in our exercise, where we have both quadratic and linear functions intersecting with each other represented in their graph forms.
Discriminant Analysis Insights
Discriminant analysis is a crucial aspect of solving quadratic equations. It determines the nature and number of solutions the equation has based on the discriminant \(b^2 - 4ac\).The discriminant provides deep insights without needing to solve the entire equation.
  • If \(b^2 - 4ac > 0\), the quadratic equation has two distinct real roots, indicating that the graph intersects the x-axis at two points.
  • If \(b^2 - 4ac = 0\), there is exactly one real root, meaning the graph touches the x-axis at one point, known as a tangent.
  • If \(b^2 - 4ac < 0\), no real roots exist, and the graph doesn't intersect the x-axis since the roots are imaginary.
For our specific problem, the discriminant was positive \(1\), indicating that there are indeed two real points of intersection for the graphs within the specified viewing rectangle.
Understanding Viewing Rectangle
The viewing rectangle in graphing is an essential concept that helps visualize specific portions of the graph. In simple terms, a viewing rectangle sets the visible range of x and y values on the graphing plane.This approach allows for easy focusing on areas of interest, especially when examining intersections.
  • The x-range of the viewing rectangle is determined by the two horizontal boundaries, such as \([-6, 2]\) in the exercise, highlighting the horizontal span of visible data.
  • The y-range is determined similarly by the vertical boundaries, as \([-5, 20]\), indicating the visible height for the y-values.
  • The viewing rectangle helps ensure computed intersections lie within the viewable area, directing attention only to relevant data points.
In our exercise, both intersections \((-4, 6)\) and \((-3, 9)\) fell within the viewing rectangle, making them valid points to consider.

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Most popular questions from this chapter

\(66-68\) Simplify the expression. $$\tan \left(\sin ^{-1} x\right)$$

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