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Chapter 4: Problem 8

The vertical line passing through the vertex of a parabola is called the _________.

Short Answer

Expert verified
axis of symmetry

Step by step solution

01

- Understanding the Parabola

A parabola is a U-shaped curve that can open either upwards or downwards. It has several key components such as the vertex, axis of symmetry, focus, and directrix.
02

- Identifying the Vertex

The vertex of a parabola is its highest or lowest point, depending on the direction of the parabola. It is located at the point \(h, k\) if the equation of the parabola is given in the vertex form \(y = a(x - h)^2 + k\).
03

- Defining the Vertical Line

The vertical line that passes through the vertex of the parabola is critical in defining its orientation.
04

- Naming the Line

This vertical line is called the 'axis of symmetry' of the parabola. It divides the parabola into two mirror-image halves.

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Key Concepts

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

Vertex
The vertex of a parabola is a crucial point that defines both its shape and position. It's the point where the parabola changes direction, either reaching a peak (if it opens downwards) or a trough (if it opens upwards). You'll often see its coordinates noted as \(h, k\). In the vertex form equation \(y = a(x - h)^2 + k\), \(h\) and \(k\) tell you exactly where on a graph the vertex is located.

Understanding the vertex helps in graphing the parabola and gives insights into the properties of the curve.
  • **Maximum or Minimum**: The vertex represents the maximum or minimum value of the parabola (depending on whether it opens downwards or upwards).
  • **Symmetry**: The vertex lies on the parabolas axis of symmetry, making it a very central and symmetric point for the curve.
Axis of Symmetry
The axis of symmetry is an imaginary vertical line passing exactly through the vertex of the parabola. This line divides the parabola into two perfect mirror images.

In mathematical terms, if the vertex of the parabola is at the point \(h, k\), then the equation of the axis of symmetry is \(x = h\). This means no matter how far you go from the vertex horizontally, both sides of the parabola will look identical.
  • **Mirror Effect**: Points that are an equal distance from the axis of symmetry are at the same height from the x-axis on both sides.
  • **Critical for Graphing**: Knowing the axis of symmetry helps in plotting points when graphing a parabola more accurately.
Focus
The focus is another essential part of a parabola located inside the curve. It lies on the axis of symmetry, and its distance from the vertex is vital for defining the parabola's shape.

In simple terms, the focus of a parabola is a point from which the distances to any point on the parabola and the directrix are equal.
  • **Reflective Property**: The focus has a unique property where any line (ray) coming from the parabola bouncing off it passes through the focus. This reflective property is used in satellite dishes and car headlights to direct signals and light.
Directrix
The directrix of a parabola is a horizontal line located opposite the focus and lies parallel to the x-axis. Unlike the focus, it is not on the parabola itself but helps shape it.

The equation of the directrix varies, but if the vertex is at \(h, k\), and the distance from the vertex to the directrix is \(p\), then the directrix will have the equation \(y = k - p\) or \(y = k + p\), depending on the parabola's orientation.
  • **Equal Distance Property**: Every point on the parabola is equidistant from the directrix and the focus.
  • **Helps in Derivation**: This property helps in deriving the parabolas standard equation.

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Most popular questions from this chapter

The price \(p\) (in dollars) and the quantity \(x\) sold of a certain product satisfy the demand equation $$ x=-6 p+600 $$ (a) Find a model that expresses the revenue \(R\) as a function of \(p .(\) Remember \(, R=x p .)\) (b) What is the domain of \(R ?\) Assume \(R\) is nonnegative. (c) What price \(p\) maximizes the revenue? (d) What is the maximum revenue? (e) How many units are sold at this price? (f) Graph \(\underline{R}\). (g) What price should the company charge to earn at least \(\$ 12,600\) in revenue?

Suppose that the quantity supplied \(S\) and the quantity demanded \(D\) of hot dogs at a baseball game are given by the following functions:$$\begin{array}{l}S(p)=-2000+3000 p \\\D(p)=10,000-1000 p\end{array}$$ where \(p\) is the price of a hot dog. (a) Find the equilibrium price for hot dogs at the baseball game. What is the equilibrium quantity? (b) Determine the prices for which quantity demanded is less than quantity supplied. (c) What do you think will eventually happen to the price of hot dogs if quantity demanded is less than quantity supplied?

(a) find the vertex and axis of symmetry of each quadratic function. (b) Determine whether the graph is concave up or concave down. (c) Graph the quadratic function. \(f(x)=2(x-6)^{2}+3\)

The following data represent the various combinations of soda and hot dogs that Yolanda can buy at a baseball game with \(\$60$$\begin{array}{|cc|}\hline \text { Soda, } s & \text { Hot Dogs, } h \\\\\hline 20 & 0 \\\15 & 3 \\\10 & 6 \\\5 & 9 \\\\\hline\end{array}$$ (a) Plot the ordered pairs \)(s, h)\( in a Cartesian plane. (b) Show that the number \)h\( of hot dogs purchased is a linear function of the number \)s\( of sodas purchased. (c) Determine the linear function that describes the relation between \)s\( and \)h$ (d) What is the domain of the linear function? (e) Graph the linear function in the Cartesian plane drawn in part (a). (f) Interpret the slope. (g) Interpret the intercepts.

The daily revenue \(R\) achieved by selling \(x\) boxes of candy is \(R(x)=9.5 x-0.04 x^{2}\). The daily cost \(C\) of selling \(x\) boxes of candy is \(C(x)=1.25 x+250 .\) (a) How many boxes of candy must the firm sell to maximize revenue? What is the maximum revenue? (b) Profit is given as \(P(x)=R(x)-C(x) .\) What is the profit function? (c) How many boxes of candy must the firm sell to maximize profit? What is the maximum profit? (d) Provide a reasonable explanation as to why the answers found in parts (a) and (c) differ. Explain why a quadratic function is a reasonable model for revenue.
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