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Chapter 8: Problem 49

Use the steps outlined in Exercise 47 to find the equation of the parabola that passes through the point (2,6) and has (1,1) as its vertex.

Short Answer

Expert verified
The equation of the parabola is \( y = 5(x-1)^2 + 1 \).

Step by step solution

01

Identify the vertex

The vertex of the parabola is given as (1,1). Therefore, in the equation y = a(x-h)^2 + k, replace h and k with 1. So the equation becomes y = a(x-1)^2 + 1.
02

Substitute the given point into the equation

We are also given that the parabola passes through the point (2,6). So, replace x and y in the equation with 2 and 6 respectively. Doing this gives us 6 = a(2-1)^2 + 1, which simplifies to 6 = a + 1.
03

Solve for 'a'

To find 'a', subtract 1 from both sides of the equation 6 = a + 1. We then obtain a = 5.
04

Write down the equation of the parabola

Now that we have found 'a', we can write the equation of the parabola. Substitute 'a', 'h', and 'k' with 5, 1, and 1 respectively into the general form of a parabola. The final equation becomes y = 5(x-1)^2 + 1.

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Key Concepts

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

Vertex Form
The vertex form of a parabola is a specific way of expressing the quadratic equation. It highlights the vertex of the parabola, which is its highest or lowest point, depending on its orientation. This form is particularly useful in coordinate geometry problems, as it makes it easier to identify the position and shape of the parabola in a graph.

The vertex form is written as \( y = a(x-h)^2 + k \), where:
  • \( a \) determines the width and the direction of the parabola (if \( a > 0 \), it opens upwards; if \( a < 0 \), it opens downwards)
  • \( (h, k) \) is the vertex of the parabola
In our exercise, by substituting \( (h, k) = (1, 1) \), we immediately place our parabola at the correct location on the graph. Understanding this form provides insight into controlling the parabola's position with simple algebraic manipulation.
Quadratic Function
A quadratic function is a polynomial function of degree two, generally expressed in the form \( y = ax^2 + bx + c \). It represents a parabolic curve on the graph. The most characteristic feature of this curve is its symmetry about a vertical axis that passes through its vertex.

In coordinate geometry, noticing the properties of the quadratic function allows us to predict how changing parameters will affect the parabola's graph:
  • The coefficient \( a \) influences the opening direction and width of the parabola. A larger absolute value of \( a \) means a tighter curve.
  • The vertex form \( y = a(x-h)^2 + k \) is a transformation of the standard form, showing precisely how to shift the parabola on the graph by changing \( h \) and \( k \).
A good understanding of quadratic functions helps in solving problems related to projectile motion, geometry, and more.
Coordinate Geometry
Coordinate geometry, also known as analytical geometry, involves using algebraic equations to describe geometric phenomena. This branch of mathematics allows us to explore the properties and relations of points, lines, and shapes through algebra.

Parabolas are common and essential forms in coordinate geometry. Using the vertex form, you can quickly determine:
  • The location of a parabola on a coordinate plane
  • The direction and opening width of the curve
Coordinate points, such as the vertex \( (h, k) \) and an additional point on the curve, help us solve for the unknown parameters of the equation, such as \( a \). In real-world applications, coordinate geometry provides the groundwork for design and engineering, where precise placement of elements is crucial.

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Most popular questions from this chapter

The following is a system of three equations in only two variables. $$\left\\{\begin{array}{r} x-y=1 \\ x+y=1 \\ 2 x-y=1 \end{array}\right.$$ (a) Graph the solution of each of these equations. (b) Is there a single point at which all three lines intersect? (c) Is there one ordered pair \((x, y)\) that satisfies all three equations? Why or why not?

Apply elementary row operations to a matrix to solve the system of equations. If there is no solution, state that the system is inconsistent. $$\left\\{\begin{array}{l}3 r+4 s-8 t=14 \\ 2 r-2 s+4 t=28\end{array}\right.$$

Consider the following system of equations. $$\left\\{\begin{array}{l}6 u+6 v-3 w=-3 \\\2 u+2 v-w=-1\end{array}\right.$$ (a) Show that each of the equations in this system is a multiple of the other equation. (b) Explain why this system of equations has infinitely many solutions. (c) Express \(w\) as an equation in \(u\) and \(v\) (d) Give two solutions of this system of equations.

A financial advisor offers three specific investment instruments: a stock- based mutual fund, a high-yield bond, and a certificate of deposit (CD). Risk factors for individual instruments can be quantified on a scale of 1 to \(5,\) with 1 being the most risky. The risk factors associated with these particular instruments are summarized in the following table.$$\begin{array}{lc} \text { Type of Investment } & \text { Risk Factor } \\ \text { Stock-based mutual fund } & 3 \\\\\text { High-yield bond } & 1 \\\\\text { CD } & 5\end{array}$$.One of the advisor's clients can tolerate an overall risk level of \(3.5 .\) In addition, the client stipulates that the amount of money invested in the mutual fund must equal the sum of the amounts invested in the high-yield bond and the CD. To satisfy the client's requirements, what percentage of the total investment should be allocated to each instrument?

This set of exercises will draw on the ideas presented in this section and your general math background. Find the inverse of $$\left[\begin{array}{lll}a & a & a \\\0 & 1 & 0 \\\0 & 0 & 1\end{array}\right]$$ where \(a\) is nonzero. Evaluate this inverse for the case in which \(a=1\)
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