# 5.1 Quadratic functions  (Page 7/15)

 Page 7 / 15

Access these online resources for additional instruction and practice with quadratic equations.

## Key equations

 general form of a quadratic function $f\left(x\right)=a{x}^{2}+bx+c$ standard form of a quadratic function $f\left(x\right)=a{\left(x-h\right)}^{2}+k$

## Key concepts

• A polynomial function of degree two is called a quadratic function.
• The graph of a quadratic function is a parabola. A parabola is a U-shaped curve that can open either up or down.
• The axis of symmetry is the vertical line passing through the vertex. The zeros, or $\text{\hspace{0.17em}}x\text{-}$ intercepts, are the points at which the parabola crosses the $\text{\hspace{0.17em}}x\text{-}$ axis. The $\text{\hspace{0.17em}}y\text{-}$ intercept is the point at which the parabola crosses the $\text{\hspace{0.17em}}y\text{-}$ axis. See [link] , [link] , and [link] .
• Quadratic functions are often written in general form. Standard or vertex form is useful to easily identify the vertex of a parabola. Either form can be written from a graph. See [link] .
• The vertex can be found from an equation representing a quadratic function. See [link] .
• The domain of a quadratic function is all real numbers. The range varies with the function. See [link] .
• A quadratic function’s minimum or maximum value is given by the $\text{\hspace{0.17em}}y\text{-}$ value of the vertex.
• The minimum or maximum value of a quadratic function can be used to determine the range of the function and to solve many kinds of real-world problems, including problems involving area and revenue. See [link] and [link] .
• The vertex and the intercepts can be identified and interpreted to solve real-world problems. See [link] .

## Verbal

Explain the advantage of writing a quadratic function in standard form.

When written in that form, the vertex can be easily identified.

How can the vertex of a parabola be used in solving real-world problems?

Explain why the condition of $\text{\hspace{0.17em}}a\ne 0\text{\hspace{0.17em}}$ is imposed in the definition of the quadratic function.

If $\text{\hspace{0.17em}}a=0\text{\hspace{0.17em}}$ then the function becomes a linear function.

What is another name for the standard form of a quadratic function?

What two algebraic methods can be used to find the horizontal intercepts of a quadratic function?

If possible, we can use factoring. Otherwise, we can use the quadratic formula.

## Algebraic

For the following exercises, rewrite the quadratic functions in standard form and give the vertex.

$f\left(x\right)={x}^{2}-12x+32$

$g\left(x\right)={x}^{2}+2x-3$

$f\left(x\right)={\left(x+1\right)}^{2}-2,\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(-1,-4\right)$

$f\left(x\right)={x}^{2}-x$

$f\left(x\right)={x}^{2}+5x-2$

$f\left(x\right)={\left(x+\frac{5}{2}\right)}^{2}-\frac{33}{4},\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(-\frac{5}{2},-\frac{33}{4}\right)$

$h\left(x\right)=2{x}^{2}+8x-10$

$k\left(x\right)=3{x}^{2}-6x-9$

$f\left(x\right)=3{\left(x-1\right)}^{2}-12,\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(1,-12\right)$

$f\left(x\right)=2{x}^{2}-6x$

$f\left(x\right)=3{x}^{2}-5x-1$

$f\left(x\right)=3{\left(x-\frac{5}{6}\right)}^{2}-\frac{37}{12},\text{\hspace{0.17em}}$ Vertex $\text{\hspace{0.17em}}\left(\frac{5}{6},-\frac{37}{12}\right)$

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.

$y\left(x\right)=2{x}^{2}+10x+12$

$f\left(x\right)=2{x}^{2}-10x+4$

Minimum is $\text{\hspace{0.17em}}-\frac{17}{2}\text{\hspace{0.17em}}$ and occurs at $\text{\hspace{0.17em}}\frac{5}{2}.\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}x=\frac{5}{2}.$

$f\left(x\right)=-{x}^{2}+4x+3$

$f\left(x\right)=4{x}^{2}+x-1$

Minimum is $\text{\hspace{0.17em}}-\frac{17}{16}\text{\hspace{0.17em}}$ and occurs at $\text{\hspace{0.17em}}-\frac{1}{8}.\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}x=-\frac{1}{8}.$

$h\left(t\right)=-4{t}^{2}+6t-1$

$f\left(x\right)=\frac{1}{2}{x}^{2}+3x+1$

Minimum is $\text{\hspace{0.17em}}-\frac{7}{2}\text{\hspace{0.17em}}$ and occurs at $\text{\hspace{0.17em}}-3.\text{\hspace{0.17em}}$ Axis of symmetry is $\text{\hspace{0.17em}}x=-3.$

$f\left(x\right)=-\frac{1}{3}{x}^{2}-2x+3$

For the following exercises, determine the domain and range of the quadratic function.

$f\left(x\right)={\left(x-3\right)}^{2}+2$

Domain is $\text{\hspace{0.17em}}\left(-\infty ,\infty \right).\text{\hspace{0.17em}}$ Range is $\text{\hspace{0.17em}}\left[2,\infty \right).$

#### Questions & Answers

A laser rangefinder is locked on a comet approaching Earth. The distance g(x), in kilometers, of the comet after x days, for x in the interval 0 to 30 days, is given by g(x)=250,000csc(π30x). Graph g(x) on the interval [0, 35]. Evaluate g(5)  and interpret the information. What is the minimum distance between the comet and Earth? When does this occur? To which constant in the equation does this correspond? Find and discuss the meaning of any vertical asymptotes.
Kaitlyn Reply
The sequence is {1,-1,1-1.....} has
amit Reply
circular region of radious
Kainat Reply
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Joel Reply
Sin(A+B) = sinBcosA+cosBsinA
Eseka Reply
Prove it
Eseka
Please prove it
Eseka
hi
Joel
June needs 45 gallons of punch. 2 different coolers. Bigger cooler is 5 times as large as smaller cooler. How many gallons in each cooler?
Arleathia Reply
7.5 and 37.5
Nando
find the sum of 28th term of the AP 3+10+17+---------
Prince Reply
I think you should say "28 terms" instead of "28th term"
Vedant
the 28th term is 175
Nando
192
Kenneth
if sequence sn is a such that sn>0 for all n and lim sn=0than prove that lim (s1 s2............ sn) ke hole power n =n
SANDESH Reply
write down the polynomial function with root 1/3,2,-3 with solution
Gift Reply
if A and B are subspaces of V prove that (A+B)/B=A/(A-B)
Pream Reply
write down the value of each of the following in surd form a)cos(-65°) b)sin(-180°)c)tan(225°)d)tan(135°)
Oroke Reply
Prove that (sinA/1-cosA - 1-cosA/sinA) (cosA/1-sinA - 1-sinA/cosA) = 4
kiruba Reply
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In a triangle ABC prove that. (b+c)cosA+(c+a)cosB+(a+b)cisC=a+b+c.
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