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Chapter 9: Problem 5

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

Short Answer

Expert verified
The vertex is (0, 0).

Step by step solution

01

Identify the Standard Form of a Parabola

A parabola can be represented in the standard form \( f(x) = ax^2 + bx + c \). In this case, the given function is \( f(x) = -3x^2 \).
02

Coefficients Identification

Identify the values of the coefficients \( a \), \( b \), and \( c \) in the given quadratic function. For \( f(x) = -3x^2 \), \( a = -3 \), \( b = 0 \), and \( c = 0 \).
03

Calculate the Vertex

The vertex of a parabola given by \( y = ax^2 + bx + c \) is located at \( x = -\frac{b}{2a} \). Substitute \( a = -3 \) and \( b = 0 \) into the formula.\[ x = -\frac{0}{2(-3)} = 0 \]
04

Find the y-coordinate

Substitute \( x = 0 \) back into the original equation to find the y-coordinate of the vertex. \[ f(0) = -3(0)^2 = 0 \].
05

State the Vertex

Combine the results. The vertex of the parabola \( f(x) = -3x^2 \) is at the point \( (0, 0) \).

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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
A parabola is a U-shaped curve and its equation can vary slightly in form. To keep things simple, the most common way to write the equation of a parabola is in what is called the standard form. The standard form of a parabola is written as \[ f(x) = ax^2 + bx + c \] Here:
  • 'a' controls the width and direction of the parabola. If 'a' is positive, the parabola opens upwards. If 'a' is negative, it opens downwards.

  • 'b' influences the position of the vertex along the x-axis.

  • 'c' moves the parabola up or down keeping the shape the same.

For example, the equation given in the exercise is \[ f(x) = -3x^2 \] Here, 'a' is -3, 'b' is 0 and 'c' is also 0. Note that since 'a' is negative (-3), the parabola will open downwards.
quadratic function
A quadratic function is a type of polynomial function and it is often written in the form: \[ f(x) = ax^2 + bx + c \] Such functions produce parabolas when graphed. They are called 'quadratic' because 'quadra' means square, and the highest degree (or power) of the variable in the function is a square (\( x^2 \)).

Key points about quadratic functions:
  • They always produce parabolas when graphed.

  • They have a squared term as the highest power of the variable.

  • The curve can either open upwards or downwards depending on the sign of 'a'.


The function \[ f(x) = -3x^2 \] is a quadratic function. Here, 'a' is -3, and since 'b' and 'c' are both 0, it simplifies the function, but this doesn't change the fact that it is a quadratic function.
vertex calculation
Finding the vertex of a quadratic function \[ f(x) = ax^2 + bx + c \] is essential, as the vertex represents the highest or lowest point on the parabola.

Here's how to calculate the vertex step by step:
  • The x-coordinate of the vertex can be found using the formula: \[ x = -\frac{b}{2a} \] In our example, we have \[ a = -3 \] and \[ b = 0 \]
  • Substitute these values into the formula: \[ x = -\frac{0}{2(-3)} = 0 \]

  • Next, substitute this x-coordinate back into the original equation to find the y-coordinate. This means we calculate:
    \[ f(0) = -3(0)^2 = 0 \]

  • Combine these results to get the coordinates of the vertex. For our example, the vertex of the parabola \[ f(x) = -3x^2 \]is at \((0, 0)\).
Understanding how to find the vertex is crucial for graphing parabolas and solving quadratic equations.

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Most popular questions from this chapter

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. $$ f(x)=\frac{-1}{x^{2}+9} $$

Use the table to evaluate each expression in parts \((a)-(d),\) if possible. (a) \((f+g)(2)\) (b) \((f-g)(4)\) (c) \((f g)(-2)\) (d) \(\left(\frac{f}{g}\right)(0)\) $$ \begin{array}{r|r|r} {}{} {x} & f(x) & g(x) \\ \hline-2 & -4 & 2 \\ \hline 0 & 8 & -1 \\ \hline 2 & 5 & 4 \\ \hline 4 & 0 & 0 \\ \hline \end{array} $$

The tables give some selected ordered pairs for functions \(f\) and \(g\). $$\begin{array}{r|r}{x} & f(x) \\\\\hline-1 & 1 \\\\\hline 2 & -1 \\\\\hline 5 & 9\end{array}$$ $$\begin{array}{c|c}x & g(x) \\\\\hline 2 & -1 \\\\\hline 7 & 5 \\\\\hline 1 & 9 \\\\\hline 9 & 20\end{array}$$ Find each of the following. $$ (g \circ f)(-1) $$

Suppose that postage rates are \(\$ 0.55\) for the first ounce, plus \(\$ 0.24\) for each additional ounce, and that each letter carries one \(\$ 0.55\) stamp and as many \(\$ 0.24\) stamps as necessary. Graph the function \(y=p(x)=\) the number of stamps on a letter weighing \(x\) ounces. Use the interval (0,5] .

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. $$ (-8,3) $$
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