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Chapter 9: Problem 6

Identify the vertex of each parabola. $$ f(x)=-4 x^{2} $$

Short Answer

Expert verified
The vertex of the parabola is \( (0, 0) \).

Step by step solution

01

Identify the general form of the quadratic function

A quadratic function is typically written in the form \[ f(x) = ax^2 + bx + c \]. In this case, the given function is \[ f(x) = -4x^2 \], which means \[ a = -4 \], \[ b = 0 \], and \[ c = 0 \].
02

Understand the vertex form of a parabola

The vertex of a parabola given by a quadratic function in the general form \[ y = ax^2 + bx + c \] can be found using the formula \[ x = -\frac{b}{2a} \] to find the x-coordinate.
03

Calculate the x-coordinate of the vertex

Use \[ x = -\frac{b}{2a} \] and substitute \[ b = 0 \] and \[ a = -4 \]. Therefore, \[ x = -\frac{0}{2(-4)} = 0 \].So, the x-coordinate of the vertex is \[ x = 0 \].
04

Determine the y-coordinate of the vertex

Substitute \[ x = 0 \] back into the original function to find the y-coordinate. \[ f(0) = -4(0)^2 = 0 \].So, the y-coordinate of the vertex is \[ y = 0 \].
05

Combine the coordinates to find the vertex

The vertex of the parabola is the point where \[ x = 0 \] and \[ y = 0 \]. Therefore, the vertex is \[ (0, 0) \].

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Key Concepts

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

quadratic function
A quadratic function is one of the simplest types of polynomial functions. It's typically expressed in the form \( f(x) = ax^2 + bx + c \). Each term in the equation corresponds to a specific part of the parabola it represents.
  • \(a\): This coefficient controls the direction and width of the parabola. If \(a\) is positive, the parabola opens upwards. If \(a\) is negative, it opens downwards. A larger absolute value of \(a\) makes the parabola narrower.
  • \(b\): This coefficient affects the location of the vertex and the symmetry of the parabola.
  • \(c\): This is the y-intercept, showing where the parabola crosses the y-axis.
For example, in the function \( f(x) = -4x^2 \), \(a = -4\), \(b = 0\), and \(c = 0\).
This results in a downward-opening parabola where the vertex will play a key role in understanding its graph.
vertex formula
The vertex of a parabola, or the highest or lowest point on its graph, is a crucial feature. To find the vertex from a quadratic function, we use the vertex formula. Here鈥檚 how it works:
  • The x-coordinate of the vertex can be found using the formula: \( x = -\frac{b}{2a} \). This is derived from completing the square or using calculus.
  • After finding the x-coordinate, substitute it back into the original function \( f(x) \) to find the y-coordinate.
Given the function \( f(x) = -4x^2 \), let's apply the formula:
1. Calculate the x-coordinate: \( x = -\frac{b}{2a} = -\frac{0}{2(-4)} = 0 \).2. Find the y-coordinate by substituting \( x = 0 \) back into the function: \( f(0) = -4(0)^2 = 0 \).
So, the vertex is at \( (0, 0) \), meaning the parabola touches the origin.
parabolic functions
Parabolic functions, often represented by quadratic equations, are a fundamental concept in algebra and calculus. They form U-shaped curves called parabolas. Here are key characteristics of parabolic functions:
  • Symmetry: Parabolas are symmetric around a vertical line called the axis of symmetry, which passes through the vertex.
  • Vertex: The highest or lowest point of the parabola. This can be a maximum (if the parabola opens downwards) or a minimum (if it opens upwards).
  • Direction: Determined by the sign of the coefficient \( a \). Positive opens up, negative opens down.
In our example \( f(x) = -4x^2 \), the parabola opens downward. Since the vertex is at \( (0,0) \), this parabola touches the origin and extends infinitely downward. Understanding these properties helps in graphing and solving quadratic functions.

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Most popular questions from this chapter

To rent a midsized car costs \(\$ 30\) per day or fraction of a day. If we pick up the car in Lansing and drop it in West Lafayette, there is a fixed \(\$ 50\) dropoff charge. Let \(C(x)\) represent the cost of renting the car for \(x\) days, taking it from Lansing to West Lafayette. Find each of the following. (a) \(C\left(\frac{3}{4}\right)\) (b) \(C\left(\frac{9}{10}\right)\) (c) \(C(1)\) (d) \(C\left(1 \frac{5}{8}\right)\) (e) \(C(2.4)\) (f) Graph \(y=C(x)\).

Compared to the graph of \(f(x)=\sqrt{x}\), the graph of \(g(x)=-2 \sqrt{x}\) is _____ across the \((x\) -axis \(/ y\) -axis \()\) and (stretched / shrunken) by a factor of ______. Some points on the graph of \(f(x)\) are \((0,0),(1,1),\) and (4,2) . Corresponding points on the graph of \(g(x)\) are a(0,0), (1, _____), and (4, ____)

Plot each point, and then use the same axes to plot the points that are symmetric to the given point with respect to the following: (a) \(x\) -axis, (b) y-axis, (c) origin. $$ (0,-3) $$

Solve each problem. See Objective 3. Suppose the population \(P\) of a certain species of fish depends on the number \(x\) (in hundreds) of a smaller kind of fish that serves as its food supply, where $$P(x)=2 x^{2}+1$$ Suppose, also, that the number \(x\) (in hundreds) of the smaller species of fish depends on the amount \(a\) (in appropriate units) of its food supply, a kind of plankton, where $$x=f(a)=3 a+2$$ Find an expression for \((P \circ f)(a),\) the relationship between the population \(P\) of the large fish and the amount \(a\) of plankton available.

Evaluate each expression. [-10.1]
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