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Chapter 8: Problem 26

(Graphing program required for part (b)). We dealt previously with systems of lines and ways to determine the coordinates of points where lines intersect. Once you know the quadratic formula, it's possible to determine where a line and a parabola, or two parabolas, intersect. As with two straight lines, at the point where the graphs of two functions intersect (if they intersect), the functions share the same \(x\) value and the same \(y\) value. a. Find the intersection of the parabola \(y=2 x^{2}-3 x+5.1\) and the line \(y=-4.3 x+10\). b. Plot both functions, labeling any intersection point(s).

Short Answer

Expert verified
The intersection points are \( (1.275, 4.5175) \) and \( (-1.925, 18.2775) \) and should be plotted for visualization.

Step by step solution

01

- Set Equations Equal

Since the intersection points of the line and the parabola will have the same x and y coordinates, start by setting the equations equal to each other: \(2x^2 - 3x + 5.1 = -4.3x + 10\).
02

- Move All Terms to One Side

Move all terms to one side to set the equation to zero: \(2x^2 - 3x + 5.1 + 4.3x - 10 = 0 \ \ 2x^2 + 1.3x - 4.9 = 0\).
03

- Apply the Quadratic Formula

The quadratic formula \(x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}\) helps to find the values of x: Here, \(a = 2\), \(b = 1.3\), and \(c = -4.9\). Calculate the discriminant first: \(b^2 - 4ac = 1.3^2 - 4 \cdot 2 \cdot (-4.9) = 1.69 + 39.2 = 40.89\). Then plug the values into the quadratic formula: \(x = \frac{-1.3 \pm \sqrt{40.89}}{4}\).
04

- Calculate the x Values

Compute the two solutions for x: \(x_1 = \frac{-1.3 + 6.4}{4} = \frac{5.1}{4} = 1.275\) and \(x_2 = \frac{-1.3 - 6.4}{4} = \frac{-7.7}{4} = -1.925\).
05

- Find Corresponding y Values

Use the line equation \(y = -4.3x + 10\) to find the y values: For \(x_1 = 1.275\): \(y = -4.3(1.275) + 10 = -5.4825 + 10 = 4.5175\). For \(x_2 = -1.925\): \(y = -4.3(-1.925) + 10 = 8.2775 + 10 = 18.2775\).
06

- Plot the Functions

Use a graphing program to plot the parabola \(y = 2x^2 - 3x + 5.1\) and the line \(y = -4.3x + 10\). Label the intersection points: \( (1.275, 4.5175) \) and \( (-1.925, 18.2775) \).

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Key Concepts

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

quadratic formula
Understanding the quadratic formula is crucial when solving systems of equations involving a parabola. The formula works for any quadratic equation of the form \[ ax^2 + bx + c = 0, \]where \(a, b, \text{and } c\) are constants. The quadratic formula is given by: \[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}. \] It provides the solutions for \(x\) by solving for the values that make the equation true.
When using the quadratic formula, it's important to calculate the discriminant first: \( b^2 - 4ac. \) The discriminant indicates the nature of the solutions:
  • If the discriminant is positive, the equation has two real and distinct solutions.
  • If it's zero, the equation has one real and repeated solution.
  • If it's negative, the equation has two complex solutions.
By plugging the appropriate values of \(a, b, \text{and } c\) into the formula, you can find the exact points where the corresponding parabola (quadratic function) intersects with another function, such as a line.
function intersection
When two functions intersect, they share the same \(x\) and \(y\) values at that point. For example, if you're looking for the intersection points between a line and a parabola, you set their equations equal to each other:
In our exercise, we set \( y = 2x^2 - 3x + 5.1 \) equal to \( y = -4.3x + 10 \). This results in the equation:
\( 2x^2 - 3x + 5.1 = -4.3x + 10.\)
After simplifying, move all terms to one side to set the equation to zero, leading to the quadratic equation. Solving this with the quadratic formula gives the \(x\) values where the functions intersect. Once we have \(x\) values, substitute back into one of the original equations to find the corresponding \(y\) values.
These \( (x, y) \) pairs are the points of intersection. In our case, the intersections were \( (1.275, 4.5175) \) and \( (-1.925, 18.2775) \). Function intersections are key in many areas such as physics, engineering, and economics, where analyzing the relationship between different quantities is essential.
graphing functions
Graphing functions provides a visual understanding of the behavior and relationship of equations. To graph the parabola \( y = 2x^2 - 3x + 5.1 \) and the line \( y = -4.3x + 10 \), follow these steps:
  • Identify key points for each function. For the parabola, determine the vertex and plot points on either side. For the line, find the points where it intersects the x and y axes.
  • Use a graphing tool or graphing software to plot precise points.
  • Draw the curves accurately to showcase how they change over their domains.
  • Clearly label the intersection points identified from solving the system of equations.
Graphing these functions not only confirms the solutions found algebraically but also provides a more intuitive understanding of the points where the functions intersect. The visual representation helps to see trends, patterns, and the overall structure of the equations they represent.

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Most popular questions from this chapter

(Graphing program optional.) The following function represents the relationship between time \(t\) (in seconds) and height \(h\) (in feet) for objects thrown upward on Pluto. For an initial velocity of \(20 \mathrm{ft} / \mathrm{sec}\) and an initial height above the ground of 25 feet, we get $$ h=-t^{2}+20 t+25 $$ a. Find the coordinates of the point where the graph intersects the \(h\) -axis. b. Find the coordinates of the vertex of the parabola. c. Sketch the graph. Label the axes. d. Interpret the vertex in terms of time and height. e. For what values of \(t\) does the mathematical model make sense?

(Graphing program required.) If an object is put in an environment with a fixed temperature \(A\) (the "ambient temperature"), then the object's temperature, \(T,\) at time \(t\) is modeled by Newton's Law of Cooling: \(T=A+C e^{-k t},\) where \(k\) is a positive constant. (Note that \(T\) is a function of \(t\) and as \(t \rightarrow+\infty,\) then \(e^{-k t} \rightarrow 0,\) so the temperature \(T\) of the object gets closer and closer to the ambient temperature, A.) A corpse is discovered in a motel room at midnight. The corpse's temperature is \(80^{\circ}\) and the room temperature is \(60^{\circ}\). Two hours later the temperature of the corpse had dropped to \(75^{\circ}\). (Problem adapted from one in the public domain site S.O.S. Math.) a. Using Newton's Law of Cooling, construct an equation to model the temperature \(T\) of the corpse over time, \(t,\) in hours since the corpse was found. b. Then determine the time of death. (Assume the normal body temperature is \(98.6^{\circ} .\) ) c. Graph the function from \(t=-5\) to \(t=5,\) and identify when the person was alive, and the coordinates where the temperature of the corpse was \(98.6^{\circ}, 80^{\circ},\) and \(75^{\circ}\).

If you do an Internet search on formulas for "ideal body weight" (IBW), one that comes up frequently was created by Dr. B. J. Devine. His formula states \(\mathrm{IBW}\) for men (in kilograms) = \(50+(2.3 \mathrm{~kg}\) per inch over 5 feet) IBW for women (in kilograms) = \(45.5+(2.3 \mathrm{~kg}\) per inch over 5 feet) a. Write the functions for IBW (in kg) for men and women, \(W_{\text {mea }}(h)\) and \(W_{\text {wamen }}(h),\) where \(h\) is a person's height in inches. Give a reasonable domain for each. b. Evaluate \(W_{\operatorname{men}}(70)\) and \(W_{\text {womes }}(66) .\) Describe your results in terms of height and weight. c. Evaluate \(W^{-1}\) mea \((77.6) .\) What does this tell you? d. Given that \(1 \mathrm{lb}=0.4356 \mathrm{~kg}\), alter the functions to create \(W_{\text {newmen }}(h)\) and \(W_{\text {acwaomen }}(h)\) so that the weight is given in pounds rather than kilograms. e. Use your functions in part (d) to find \(W^{-1}\) newwomea (125) . What does this tell you?

A stone is dropped into a pond, causing a circular ripple that is expanding at a rate of \(13 \mathrm{ft} / \mathrm{sec}\). Describe the area of the circle as a function of time.

A quadratic function has two complex roots, \(r_{1}=1+i\) and \(r_{2}=1-i\). Use the Factor Theorem to find the equation of this quadratic, assuming \(a=1\), and then put it into standard form.
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