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Chapter 9: Problem 7

Identify the vertex of each parabola. $$ f(x)=\frac{1}{3} x^{2} $$

Short Answer

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

Step by step solution

01

Identify the Standard Form

The standard form of a parabola is \[ f(x) = ax^2 + bx + c \]. In this case, the function given is \[ f(x) = \frac{1}{3} x^2 \]. Here, \[ a = \frac{1}{3} \], \[ b = 0 \], and \[ c = 0 \].
02

Use the Formula for the Vertex

The vertex of a parabola given by \[ f(x) = ax^2 + bx + c \] can be found using the formula \[ x = -\frac{b}{2a} \].
03

Calculate the X-Coordinate

Substitute \[ b = 0 \] and \[ a = \frac{1}{3} \] into the formula: \[ x = -\frac{0}{2 \cdot \frac{1}{3}} = 0 \]. Therefore, the x-coordinate of the vertex is \[ 0 \].
04

Calculate the Y-Coordinate

Substitute \[ x = 0 \] back into the original function to find the y-coordinate: \[ f(0) = \frac{1}{3} (0)^2 = 0 \]. Therefore, the y-coordinate is \[ 0 \].
05

State the Vertex

Combining the x and y coordinates, the vertex of the parabola \[ f(x) = \frac{1}{3} x^2 \] is \[ (0, 0) \].

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Key Concepts

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

Standard Form of a Parabola
To understand parabolas, first, we need to know their standard form. A quadratic function typically describes a parabola and is written as \[ f(x) = ax^2 + bx + c \].
    \t
  • \( a \) determines the parabola's direction and shape.
    • \t
  • \( b \) affects the position of the vertex along the x-axis.
    • \t
  • \( c \) denotes the y-intercept.
In the quadratic function from the given exercise, \[ f(x) = \frac{1}{3} x^2 \], we see that \( a = \frac{1}{3} \), \( b = 0 \), and \( c = 0 \). This simplified equation shows a parabola opening upwards (since \( a \) is positive) and centered at the origin (0, 0). This clear representation helps us identify the crucial components needed to find the vertex.
The Vertex Formula
Next, let's dive into how we use the vertex formula. The vertex is a critical point on the parabola鈥攊t represents the maximum or minimum point, depending on the parabola's direction. For any parabola described by \[ f(x) = ax^2 + bx + c \],the x-coordinate of the vertex can be found using \[ x = -\frac{b}{2a} \].Let鈥檚 break it down:
    \t
  • The numerator \( -b \) indicates a reflection across the x-axis.
    • \t
  • The denominator \( 2a \) adjusts this value based on the parabola's width and direction.
For our function: \( b = 0 \) and \( a = \frac{1}{3} \). Substituting these values in:\[ x = -\frac{0}{2 \cdot \frac{1}{3}} = 0 \]. This tells us that the vertex is centered along the y-axis at \( x = 0 \).
Quadratic Function
Finally, let鈥檚 understand quadratic functions in more general terms. A quadratic function is any function that can be plotted as a parabola. It is essential to recognize that:
    \t
  • Any change in \( a \) affects the parabola's width and whether it opens upwards or downwards (positive \( a \) means upwards).
    • \t
  • Any non-zero \( b \) shifts the vertex along the x-axis.
    • \t
  • \( c \) moves the entire graph up or down the y-axis, indicating where the parabola crosses the y-axis.
Given \( f(x) = \frac{1}{3} x^2 \), the quadratic equation represents a very straightforward case:
    \t
  • The vertex at \( (0, 0) \)
    • \t
  • No horizontal or vertical shifts since \( b = 0 \) and \( c = 0 \)
    • \t
  • An upwards opening parabola that is relatively wide due to \( a = \frac{1}{3} \).
Understanding these variations helps visualize how different quadratic functions behave when plotted.

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Most popular questions from this chapter

The graph shows how Social Security trust fund assets are expected to change and suggests that a quadratic function would be a good fit to the data. The data are approximated by the function $$ f(x)=-20.57 x^{2}+758.9 x-3140 . $$ In the model, \(x=10\) represents \(2010, x=15\) represents \(2015,\) and so on, and \(f(x)\) is in billions of dollars. Algebraically determine the vertex of the graph. Express the \(x\) -coordinate to the nearest hundredth and the \(y\) -coordinate to the nearest whole number. Interpret the answer as it applies to this application.

Use the tests for symmetry to decide whether the graph of each relation is symmetric with respect to the \(x\) -axis, the y-axis, or the origin. More than one of these symmetries, or none of them, may apply. $$ y=x^{3} $$

The snow depth in a particular location varies throughout the winter. In a typical winter, the snow depth in inches might be approximated by the following function. $$ f(x)=\left\\{\begin{array}{ll} 6.5 x & \text { if } 0 \leq x \leq 4 \\ -5.5 x+48 & \text { if } 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. $$ (0,-3) $$

Evaluate each expression. [-5]
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