Problem 39 For each quadratic function, (a)... [FREE SOLUTION]

91影视

Intermediate Algebra : Concepts and Applications

Marvin L. Bittinger, David J. Ellenbogen, Barbara L. Johnson

$Math Studyset 91影视 Explanations$ Math

10 Edition

Chapter 8: Problem 39

For each quadratic function, (a) find the vertex, the axis of symmetry, and the maximum or minimum function value and (b) graph the function. $$f(x)=-3 x^{2}+5 x-2$$

Short Answer

Expert verified

Vertex: $\left(\frac{5}{6}, \frac{1}{12}\right)$. Axis of symmetry: $x=\frac{5}{6}$. Maximum value: $\frac{1}{12}.$

Step by step solution

Identify the coefficients

Given the quadratic function $f(x)=-3x^2+5x-2$, identify the coefficients: $a=-3$, $b=5$, and $c=-2$.

Calculate the vertex

The vertex of a quadratic function in the form of $ax^2+bx+c$ can be found using the formula $x=-\frac{b}{2a}$. Substitute $b=5$ and $a=-3$: $x=-\frac{5}{2(-3)}=-\frac{5}{-6}=\frac{5}{6}$. To find the y-coordinate of the vertex, substitute $x=\frac{5}{6}$ into $f(x)$: $f\left(\frac{5}{6}\right)=-3\left(\frac{5}{6}\right)^2+5\left(\frac{5}{6}\right)-2 = -3\cdot\frac{25}{36}+\frac{25}{6}-2=\frac{-75}{36}+\frac{150}{36}-\frac{72}{36}=\frac{3}{36}=\frac{1}{12}.$ So, the vertex is $\left(\frac{5}{6}, \frac{1}{12}\right).$

Determine the axis of symmetry

The axis of symmetry of a parabola is the vertical line that passes through the vertex. It has the equation $x=\frac{5}{6}$.

Find the maximum or minimum function value

Since $a=-3<0$, the parabola opens downwards, and the vertex represents the maximum point. Thus, the maximum function value is the y-coordinate of the vertex, which is $\frac{1}{12}.$

Graph the function

To graph $f(x)=-3x^2+5x-2$, plot the vertex $\left(\frac{5}{6}, \frac{1}{12}\right)$ and draw the axis of symmetry $x=\frac{5}{6}$. Then, plot additional points on either side of the vertex and sketch the parabola opening downwards.

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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 a type of polynomial function with a degree of 2. Its general form is given by $ f(x) = ax^2 + bx + c $, where:

$ a $ is the coefficient of $ x^2 $,
$ b $ is the coefficient of $ x $, and
$ c $ is the constant term.

In this exercise, the quadratic function is $ f(x) = -3x^2 + 5x - 2 $, where $ a = -3 $, $ b = 5 $, and $ c = -2 $.
Quadratic functions typically graph to a U-shaped curve known as a parabola.
The direction of the parabola depends on the coefficient $ a $. If $ a \gt 0 $, the parabola opens upwards. If $ a \lt 0 $, it opens downwards.
For our function, since $ a = -3 \lt 0 $, the parabola opens downwards.

vertex formula

The vertex of a parabola is a crucial point that gives us the maximum or minimum value of the quadratic function.
It can be found using the vertex formula: $ x = -\frac{b}{2a} $.
This formula derives the x-coordinate of the vertex. To find the y-coordinate, substitute $ x $ into $ f(x) $.
In this exercise, we have: $ a = -3 $ and $ b = 5 $.
Using the vertex formula: $ x = -\frac{5}{2(-3)} = \frac{5}{6} $.
Substitute $ x = \frac{5}{6} $ into $ f(x) $:\[ f\left( \frac{5}{6} \right) = -3\left( \frac{5}{6} \right)^2 + 5\left( \frac{5}{6} \right) - 2 = \frac{1}{12} \]
Therefore, the vertex is $ \left( \frac{5}{6}, \frac{1}{12} \right) $.

parabola symmetry

A parabola exhibits symmetry around its axis of symmetry.
This axis is a vertical line that passes through the vertex.
The equation for the axis of symmetry comes from the x-coordinate of the vertex.
In this exercise, that is $ x = \frac{5}{6} $.
This line divides the parabola into two mirror-image halves.
Understanding the axis of symmetry helps us graph the parabola more accurately.
Any point on one side of the axis has a corresponding point equidistant on the other side.

maximum and minimum values

The vertex of a quadratic function provides the maximum or minimum value of the function, known as the vertex value.
If the parabola opens upward ($ a \gt 0 $), the vertex represents the minimum value.
If the parabola opens downward ($ a \lt 0 $), the vertex is the maximum value.
In this exercise, since $ a = -3 \lt 0 $, the parabola opens downward, indicating a maximum value.
The maximum value is given by the y-coordinate of the vertex, which is $ \frac{1}{12} $.
Therefore, the maximum function value is $ \frac{1}{12} $.

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91影视

For each quadratic function, (a) find the vertex, the axis of symmetry, and the maximum or minimum function value and (b) graph the function. $$f(x)=-3 x^{2}+5 x-2$$

Short Answer

Step by step solution

Identify the coefficients

Calculate the vertex

Determine the axis of symmetry

Find the maximum or minimum function value

Graph the function

Key Concepts

quadratic function

vertex formula

parabola symmetry

maximum and minimum values

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