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

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 $\text{\hspace{0.17em}}\left[2,\infty \right).$

how do I set up the problem?
what is a solution set?
Harshika
hello, I am happy to help!
Abdullahi
find the value of 2x=32
divide by 2 on each side of the equal sign to solve for x
corri
X=16
Michael
Want to review on complex number 1.What are complex number 2.How to solve complex number problems.
Beyan
use the y -intercept and slope to sketch the graph of the equation y=6x
how do we prove the quadratic formular
hello, if you have a question about Algebra 2. I may be able to help. I am an Algebra 2 Teacher
thank you help me with how to prove the quadratic equation
Seidu
may God blessed u for that. Please I want u to help me in sets.
Opoku
what is math number
4
Trista
x-2y+3z=-3 2x-y+z=7 -x+3y-z=6
Need help solving this problem (2/7)^-2
x+2y-z=7
Sidiki
what is the coefficient of -4×
-1
Shedrak
the operation * is x * y =x + y/ 1+(x × y) show if the operation is commutative if x × y is not equal to -1
An investment account was opened with an initial deposit of \$9,600 and earns 7.4% interest, compounded continuously. How much will the account be worth after 15 years?
lim x to infinity e^1-e^-1/log(1+x)
given eccentricity and a point find the equiation