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Quadratic Functions

Quadratic functions are sometimes considered as functions, or members of the exponential family of functions. The functions are defined in terms of the roots of the quadratic equation, ax2 + bx + c = 0, provided that b2 – 4ac is not 0. If it is 0, then the functions are not defined, but if it is not 0, they are defined by where a, b, and c are as defined above.

Quadratic Functions

621 Practice Problems
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02:08
Introductory and Intermediate Algebra for College Students

Write the equation of each parabola in $f(x)=a(x-h)^{2}+k$ form.
Vertex: (-3,-4)$;$ The graph passes through the point (1,4)

Quadratic Equations and Functions
Quadratic Functions and Their Graphs
01:45
Introductory and Intermediate Algebra for College Students

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
The graph of $f(x)=-2(x+4)^{2}-8$ has one $y$ -intercept and two $x$ -intercepts.

Quadratic Equations and Functions
Quadratic Functions and Their Graphs
01:21
Introductory and Intermediate Algebra for College Students

Determine whether each statement is true or false. If the statement is false, make the necessary change(s) to produce a true statement.
No quadratic functions have a range of $(-\infty, \infty)$

Quadratic Equations and Functions
Quadratic Functions and Their Graphs

Using Quadratic Functions as Models

145 Practice Problems
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05:07
Algebra and Trigonometry

A farmer finds that if she plants 75 trees per acre, each tree will yield 20 bushels of fruit. She estimates that for each additional tree planted per acre, the yield of each tree will decrease by 3 bushels. How many trees should she plant per acre to maximize her harvest?

Polynomial and Rational Functions
Quadratic Functions
Sheryl Ezze
03:09
Algebra and Trigonometry

For the following exercises, determine whether there is a minimum or maximum value to each quadratic function. Find the value and the axis of symmetry.
$$h(t)=-4 t^{2}+6 t-1$$

Polynomial and Rational Functions
Quadratic Functions
Chelsi Boswell
01:23
Calculus and Its Applications

The quantity sold $x$ of a plasma television is inversely proportional to the price $p$. If 85,000 plasma TVs sold for $\$ 2900$ each, how many will be sold if the price is $\$ 850$ each?

Functions, Graphs, and Models
Nonlinear Functions and Models
Carson Merrill

Using Quadratic Models to Represent Data

56 Practice Problems
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04:48
Elementary Statistics

Listed below are mean amounts of carbon dioxide concentrations (parts per million) in our atmosphere for each decade, beginning with the 1880 s. Find the best model and then predict the value for $2090-2099 .$ Comment on the result.
$$\begin{array}{rrrrrrrr}292 & 294 & 297 & 300 & 304 & 307 & 309 & 314 & 320 & 331 & 345 & 360 & 377\end{array}$$

Correlation and Regression
Nonlinear Regression
James Kiss
02:21
Elementary Statistics

According to Benford's law, a variety of different data sets include numbers with leading (first) digits that occur with the proportions listed in the following table.
$$\begin{array}{l|c|c|c|c|c|c|c|c|c}\hline \text { Leading Digit } & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 & 9 \\\hline \text { Proportion } & 0.301 & 0.176 & 0.125 & 0.097 & 0.079 & 0.067 & 0.058 & 0.051 & 0.046 \\\hline\end{array}$$

Correlation and Regression
Nonlinear Regression
James Kiss
01:31
Elementary Statistics

The table lists intensities of sounds as multiples of a basic reference sound. A scale similar to the decibel scale is used to measure the sound intensity.
$$\begin{array}{l|l|l|l|l|l}\hline \text { Sound Intensity } & 316 & 500 & 750 & 2000 & 5000 \\\hline \text { Scale Value } & 25.0 & 27.0 & 28.75 & 33.0 & 37.0 \\\hline\end{array}$$

Correlation and Regression
Nonlinear Regression
James Kiss

Quadratic Inequalities

109 Practice Problems
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01:49
Precalculus : Building Concepts and Connections

Solve the quadratic inequality.
$$x^{2} \leq 4$$

Polynomial and Rational Functions
Polynomial and Rational Inequalities
02:53
Precalculus : Building Concepts and Connections

Solve the inequality algebraically or graphically.
$$x^{2}-x+1 \geq 0$$

More About Functions and Equations
Quadratic Inequalities
Aamir Mithaiwala
02:30
Precalculus : Building Concepts and Connections

Solve the inequality algebraically or graphically.
$$2 x^{2}-3 x<1$$

More About Functions and Equations
Quadratic Inequalities
Aamir Mithaiwala

The Square Root Property and Completing the Square

34 Practice Problems
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02:28
Prealgebra and Introductory Algebra

For a quadratic equation of the form $x^{2}+b x+c=0,$ the sum of the solutions is equal to the opposite of $b$, and the product of the solutions is equal to $c .$ For example, the solutions of the equation $x^{2}+5 x+6=0$ are $-2$ and $-3 .$ The sum of the solutions is
$-5,$ the opposite of the coefficient of $x$. The product of the solutions is $6,$ the constant term. This is one way to check the solutions of a quadratic equation. Use this method to determine whether the given numbers are solutions of the equation. If they are not solutions of the equation, find the solutions.
$$x^{2}-8 x-14=0 ;-4+\sqrt{15} \text { and }-4-\sqrt{15}$$

Quadratic Equations
Solving Quadratic Equations by Using the Quadratic Formula
Ashley Jordon
00:43
Prealgebra and Introductory Algebra

Explain why the equation $(x-2)^{2}=-4$ does not have a real number solution.

Quadratic Equations
Solving Quadratic Equations by Completing the Square
Jonathon Brumley
02:02
Prealgebra and Introductory Algebra

For Exercises 56 to $58,$ solve.
$$\sqrt{2 x+7}-4=x$$

Quadratic Equations
Solving Quadratic Equations by Completing the Square
Jonathon Brumley

The Quadratic Formula

16 Practice Problems
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01:24
Prealgebra and Introductory Algebra

Factoring, completing the square, and using the quadratic formula are three methods of solving quadratic equations. Describe each method, and cite the advantages and disadvantages of each.

Quadratic Equations
Solving Quadratic Equations by Using the Quadratic Formula
Ashley Jordon
01:56
Prealgebra and Introductory Algebra

Solve by using the quadratic formula. Approximate the solutions to the nearest thousandth.
$$5 v^{2}-v-5=0$$

Quadratic Equations
Solving Quadratic Equations by Using the Quadratic Formula
Ashley Jordon
00:33
Prealgebra and Introductory Algebra

True or false? If you use the quadratic formula to solve $a x^{2}+b x+c=0$ and get rational solutions, then you could have solved the equation by factoring.

Quadratic Equations
Solving Quadratic Equations by Using the Quadratic Formula
Ashley Jordon

Quadratic Functions and Their Graphs

8 Practice Problems
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01:14
Prealgebra and Introductory Algebra

Determine the $x$ - and $y$ -intercepts.
$$y=x^{2}+5 x-6$$

Quadratic Equations
Graphing Quadratic Equations in Two Variables
Brittany Scott
01:28
Prealgebra and Introductory Algebra

Graph.
(GRAPH CANNOT COPY)
$$y=x^{2}-4 x$$

Quadratic Equations
Graphing Quadratic Equations in Two Variables
Brittany Scott
00:46
Prealgebra and Introductory Algebra

Evaluate the function for the given value of $x$.
$$f(x)=-2 x^{2}+2 x-1 ; x=-3$$

Quadratic Equations
Graphing Quadratic Equations in Two Variables
Brittany Scott

Quadratic and Other Nonlinear Inequalities

5 Practice Problems
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03:30
Beginning and Intermediate Algebra

Solve each quadratic inequality. Graph the solution set and write the solution in interval notation.
$$c^{2}+5 c<36$$

Conic Sections, Nonlinear Inequalities, and Nonlinear Systems
Quadratic and Rational Inequalities
Ernest Castorena
00:27
Intermediate Algebra

A. Solve: $x^{2}-x-12>0$
B. Find a rational inequality in one variable that has the same solution set as the quadratic inequality in part (a).

Quadratic Equations, Functions, and Inequalities
Quadratic and Other Nonlinear Inequalities
Brittany Scott
01:47
Intermediate Algebra

The number of people $n$ in a mall is modeled by the formula
$$
n=-100 x^{2}+1,200 x
$$
where $x$ is the number of hours since the mall opened. If the mall opened at 9 A.M., when were there $2,000$ or more people in it?

Quadratic Equations, Functions, and Inequalities
Quadratic and Other Nonlinear Inequalities
Sanchit Gogia

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