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Chapter 10: Problem 10

Write an equation in general form for the parabola shown, with \(x\)-intercepts \(-1\) and \(2.5\) and \(y\)-intercept \(-5\).

Short Answer

Expert verified
The equation of the parabola is \(y = 2x^2 - 3x - 5\).

Step by step solution

01

Understand the Parabola Form

The general form of a parabola is given by the equation \[ ax^2 + bx + c = 0 \]We will convert the information of intercepts into this form.
02

Use the Intercept Form

Interceptions on the \(x\)-axis allow us to use the factored form of a parabola: \[ y = a(x - x_1)(x - x_2) \]where \(x_1 = -1\) and \(x_2 = 2.5\) are the \(x\)-intercepts. Therefore, the equation becomes:\[ y = a(x + 1)(x - 2.5) \]
03

Expand the Factored Form

Expand \((x + 1)(x - 2.5)\):\[ y = a(x^2 - 2.5x + x - 2.5) = a(x^2 - 1.5x - 2.5) \]
04

Use the Given \(y\)-intercept

Since the \(y\)-intercept is \(-5\), substitute \(x = 0\) and \(y = -5\):\[ -5 = a(0^2 - 1.5(0) - 2.5) \]This simplifies to:\[ -5 = a(-2.5) \]Solving for \(a\), we get:\[ a = \frac{-5}{-2.5} = 2 \]
05

Write the Final Equation

Using \(a = 2\) in the expanded equation:\[ y = 2(x^2 - 1.5x - 2.5) \]Distribute \(2\):\[ y = 2x^2 - 3x - 5 \]

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Key Concepts

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

Parabolas
A parabola is a U-shaped curve on a graph. It is the graphical representation of a quadratic equation. Parabolas have several key properties that are useful to understand:
  • They can open upwards or downwards.
  • The vertex is the highest or lowest point on the parabola.
  • The axis of symmetry is a vertical line that passes through the vertex and divides the parabola into two mirror-image halves.
  • The direction the parabola opens is determined by the sign of the coefficient of the squared term; if positive, it opens upwards, if negative, downwards.
In this exercise, we worked with a parabola whose equation was determined by its intercepts. Recognizing the properties and shape of a parabola is crucial to forming and solving quadratic equations effectively.
X-Intercepts
X-intercepts are the points where a parabola crosses the x-axis. They are crucial in forming quadratic equations in factored form. For a parabola described by \[ ax^2 + bx + c = 0 \]the x-intercepts can be determined by setting \( y = 0 \) and solving for \( x \).
These intercepts help in rewriting the equation in factored form:
  • If \( x_1 \) and \( x_2 \) are the x-intercepts, the equation in factored form is \( y = a(x - x_1)(x - x_2) \).
  • In our exercise, the x-intercepts were -1 and 2.5, which gives us the factored form \( y = a(x + 1)(x - 2.5) \).
Understanding x-intercepts lets us easily transition to other forms of quadratic equations.
Y-Intercepts
The y-intercept of a parabola is the point where it crosses the y-axis. This information helps identify how the parabola behaves when \( x = 0 \).
The y-intercept is found by substituting \( x = 0 \) in the quadratic equation \[ ax^2 + bx + c \],and solving for \( y \).
In this problem, the y-intercept was given as -5. This value allows you to solve for any unknown coefficients in the factored form of a quadratic equation:
  • Set the equation to \( y = a(x + 1)(x - 2.5) \).
  • Substitute \( x = 0 \) and \( y = -5 \), then solve for \( a \).
  • Apply that \( a \) value to find the full quadratic equation in standard form.
Understanding y-intercepts gives you a complete grasp of a parabola's graph.
Factored Form
Factored form is a way of writing quadratic equations that highlight the x-intercepts straightaway. It is particularly useful for graphing, as it clearly shows where the curve will cross the x-axis.
The general expression for the factored form is given by\[ y = a(x - x_1)(x - x_2) \].
It breaks down the parabola into the easily identifiable intercept points:
  • The variables \( x_1 \) and \( x_2 \) directly represent the x-intercepts.
  • Working off these intercept points allows the equation to be expanded to the standard quadratic form \( ax^2 + bx + c \) by foil expansion.
In our exercise, it helped us quickly pinpoint how the parabola interacts with the axes and allowed us to transition smoothly into the general form.

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Most popular questions from this chapter

Draw and label a segment like this one. Plot and label points on your segment to represent the probability for each situation. a. You will eat breakfast tomorrow morning. b. It will rain or snow sometime during the next month in your hometown. c. You will be absent from school fewer than five days this school year. d. You will get an A on your next mathematics test. e. The next person to walk in the door will be under 30 years old. f. Next Monday every teacher at your school will give 100 free points to each student. g. Earth will rotate once on its axis in the next 24 hours.

In April 2004, the faculty at Princeton University voted that each department could give A grades to no more than \(35 \%\) of their students. Japanese teacher Kyoko Loetscher felt that 11 of her 20 students deserved A's, as they had earned better than \(90 \%\) in the course. However, she could give A's to only \(35 \%\) of her students. How many students is this? Draw two relative frequency circle graphs: one that shows the grades (A's versus non-A's) that Loetscher would like to give and one that shows the grades she is allowed to give. (Newsweek, Febnary 14, 2005, p. 8) (a)

A product like \(3 \cdot 2 \cdot 1\) or \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1\) is called a factorial expression and is written with an exclamation point, like this: \(3 \cdot 2 \cdot 1=3\) ! and \(5 \cdot 4 \cdot 3 \cdot 2 \cdot 1=5\) !. a. How can you calculate 8 !? b. How can you use factorial notation to calculate the number of permutations of 10 objects chosen 10 at a time? \([-\square\) See Calculator Note \(10 \mathrm{~F}\) to leam how to compute \(n\) with your calculator. 4 ] c. Write an expression in factorial notation that can be used to calculate \({ }_{n} P_{n}\).

You have purchased 4 tickets to a school music department raffle. Three prizes will be awarded, and 150 tickets were sold. a. How many ways can the three prizes be assigned to the 150 tickets if the prizes are different? (a) b. How many ways can the three prizes be assigned to the 150 tickets if the prizes are the same?

In a random walk, you move according to rules with each move being determined by a random process. The simplest type of random walk is a one-dimensional walk where each move is either one step forward or one step backward on a number line. a. Start at 0 on the number line and flip a coin to determine your move. Heads means you take one step forward to the next integer, and tails means you take one step backward to the previous integer. What sequence of six tosses will land you on the number-line locations \(+1,+2,+1,+2,+3,+2\) ? (a) b. Explore a one-dimensional walk of 100 moves using a calculator routine that randomly generates \(+1\) or \(-1\). In list \(\mathrm{L} 1\), generate random numbers with 1 representing a step forward and \(-1\) representing a step backward. Describe what you need to do with list L1 to show your number-line location after every step. [r \([\) See Calculator Note 10A and 10B. 4] c. Describe the results of your simulation. Is this what you expected?
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