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A type of function that commonly appears in various day-to-day situations is the *quadratic function*. This lesson will define, examine characteristics, and provide examples of such functions. Enjoy!
### Catch-Up and Review

**Here are a few recommended readings before getting started with this lesson.**

Explore

Consider four different graphs of functions. Examine them closely to determine both similarities and differences.

Discussion

Concept

A quadratic expression is a specific type of polynomial that follows the form $ax_{2}+bx+c.$ Each term has a specific name.

For an expression in this form to be quadratic, $a$ must be a non-zero number. Consider a few examples of quadratic functions.

Quadratic Expression |
---|

$x_{2}+3x+5$ |

$5x_{2}+41 x+23$ |

$53 x_{2}+21 x+4.3$ |

Example

Dylan and his crew are traveling to a newly discovered planet named Harmonica where life is possible. The gravity of different space objects cause their space shuttle to follow a curved path.
Combine like terms to determine whether this is a quadratic expression.
Combine like terms to determine whether this is a quadratic expression. ### Hint

### Solution

Notice that there are multiple combinations of like terms. Simplify this expression by combining the like terms.
The highest degree of the expression is $2.$ This means that this is a quadratic expression.
Notice that there are multiple combinations of like terms. Simplify this expression by combining the like terms.
The highest degree of the expression is $3.$ This means that this is **not** a quadratic expression.

External credits: @catalyststuff

a The path can be represented by the following expression.

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b During their flight to Harmonica, Dylan and the crew witnessed a spectacular comet named Celestial Wanderlust. Its path is described by the following expression.

{"type":"choice","form":{"alts":[" Quadratic expression","Non-quadratic expression"],"noSort":true},"formTextBefore":"","formTextAfter":"","answer":1}

a Simplify the expression by combining like terms.

b Simplify the expression by combining like terms.

a Consider the given algebraic expression.

$3x_{2}+5x−6x_{3}+x_{2}+14+2x_{3}−11x+4x_{3}$

CommutativePropAdd

Commutative Property of Addition

$3x_{2}+x_{2}+5x−11x−6x_{3}+2x_{3}+4x_{3}+14$

AddSubTerms

Add and subtract terms

$4x_{2}−6x−6x_{3}+6x_{3}+14$

CommutativePropAdd

Commutative Property of Addition

$4x_{2}−6x+6x_{3}−6x_{3}+14$

DiffZero

$6x_{3}−6x_{3}=0$

$4x_{2}−6x+0+14$

IdPropAdd

Identity Property of Addition

$4x_{2}−6x+14$

b Consider the given algebraic expression.

$2+7x_{3}−5x_{2}−6x+3x_{3}+6x_{2}+9+4x$

CommutativePropAdd

Commutative Property of Addition

$2+9+7x_{3}+3x_{3}−5x_{2}+6x_{2}−6x+4x$

AddSubTerms

Add and subtract terms

$11+10x_{3}+x_{2}−2x$

CommutativePropAdd

Commutative Property of Addition

$10x_{3}+x_{2}−2x+11$

Pop Quiz

Consider the given algebraic expression. Determine whether it is a quadratic expression by combining like terms.

Discussion

A parabola is a curve that is geometrically defined as the locus of all points equidistant from a line and a point not on the line. The line is called the directrix, and the point is the focus of the parabola. In the following applet, point $P$ is equidistant from the directrix $d$ and the focus $F.$

A parabola can be *vertical* or *horizontal*. A vertical parabola can open upward or downward. In comparison, a horizontal parabola can open to the left or the right. The graph of a quadratic function is a vertical parabola.

Discussion

A parabola either opens upward or downward. This is the direction of the parabola . If the leading coefficient $a$ of the corresponding equation is positive, the parabola opens upward. If the coefficient is negative, the parabola opens downward.

Discussion

A quadratic function is a polynomial function of degree $2$ that can always be written in the form $y=ax_{2}+bx+c,$ with $a =0.$ Notice that the highest exponent of the independent variable is $2.$ The graph of any quadratic function is a vertical parabola.

Pop Quiz

Consider the graph of a function. Identify whether the graph shows a parabola or not.

Pop Quiz

Consider the graph of a quadratic function. Is the leading coefficient of the parabola positive or negative?

Example

Once Dylan and his crew arrived to Harmonica, they started settling down and building a city. They are planning to build a bridge over the River of Crystallinity that will have shape of a parabola.
Use the table of values to graph the shape of the bridge. Note that $x$ represents the distance from the base of one side of the bridge, while $y$ represents the height of the bridge over the ground.
### Answer

### Hint

### Solution

External credits: @rawpixel.com

Each pair of numbers in the table of values is an ordered pair that represents the coordinates of a point on the graph. Plot all the points and then connect them with a smooth curve.

The following table of values is given for the bridge in the shape of a parabola. There, $x$ represents the distance from the base of one side of the bridge, while $y$ represent the height of the bridge over the ground.

$x$ | $y$ |
---|---|

$0$ | $-0.6$ |

$1$ | $0.5$ |

$2$ | $1.4$ |

$4$ | $2.6$ |

$6$ | $3$ |

$8$ | $2.6$ |

$10$ | $1.4$ |

$12$ | $-0.6$ |

Note that each pair of values of $x$ and $y$ forms an ordered pair $(x,y).$ These ordered pairs are coordinates of the points on the parabola. Start by plotting them on a coordinate plane.

Now, connect the points with a smooth curve. This way the graph of the parabola is drawn.

Example

The crew also brought some species of animals from the Earth. They started observing the changes in their populations over time. Here is the table of values that shows the changes in kangaroo population over the period of one year.

Use the table of values to graph the population growth of kangaroos. Note that $x$ represents the number of month since the observation began and $y$ represents the population of kangaroos in the thousands.

The following table of values represents the population of kangaroos over one year. Here, $x$ represents the number of month since the observation began and $y$ represents the population of kangaroos in thousands.

$x$ | $y$ |
---|---|

$0$ | $30$ |

$2$ | $20$ |

$4$ | $14$ |

$6$ | $12$ |

$8$ | $14$ |

$10$ | $20$ |

$12$ | $30$ |

Note that each pair of values of $x$ and $y$ forms an ordered pair $(x,y).$ These ordered pairs are coordinates of the points on the parabola. Start by plotting them on a coordinate plane.

Now, connect the points with a smooth curve. This way the graph of the parabola is obtained.

Example

Another important aspect of settling into life on a new planet is setting up a communication system with Earth. Kevin and his crew began working on installing satellite dishes and a few telescopes. The number of communication devises $y$ on different days $x$ since the work has begun is given by the following table of values.

### Hint

Each pair of numbers in the table of values is an ordered pair that represents the coordinates of a point on the graph. Plot all the points and then connect them with a smooth curve.

### Solution

Use the table of values to graph the function that represents the number of communication devices on Harmonica. How many communication devices will there be on Harmonica on Day $7?$ Round the number to the closest tenth.

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Consider the table of values that represents the total number of communication devices $y$ on Harmonica after $x$ days of working on this task.

$x$ | $y$ |
---|---|

$0$ | $0$ |

$1$ | $0.8$ |

$2$ | $3.2$ |

$3$ | $7.2$ |

$4$ | $12.8$ |

$5$ | $20$ |

$6$ | $28.8$ |

Note that each pair of values of $x$ and $y$ forms an ordered pair $(x,y).$ These ordered pairs are coordinates of the points on the parabola. Start by plotting them on a coordinate plane.

Now, connect the points with a smooth curve. This way the graph of the parabola is obtained.

Finally, use the graph to determine how many communication devices there will be on Harmonica on Day $7.$ Draw an arrow from $x=7$ to the parabola and then from the parabola to the $y-$axis.

Therefore, there will be about $40$ communication devices on Day $7.$

Closure

This lesson focused on vertical parabolas, their characteristics and some examples. With that covered, what about horizontal parabolas? Consider a few examples.
**not** a function. This is true for *all* horizontal parabolas.

Do horizontal parabolas also represent quadratic functions? That is determined by performing a vertical line test. Recall that a relation is not a function if there are multiple $y-$values corresponding to the same $x-$value.

Since the vertical parabola does not pass the vertical line test, it is

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