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{{ printedBook.courseTrack.name }} {{ printedBook.name }} A quadratic function is a polynomial function of degree $2.$ That means that the highest exponent of the independent variable is $2.$ The standard form of a quadratic function is expressed as follows.

$y=ax_{2}+bx+c$

Here, $a,b,$ and $c$ are real numbers and $a =0.$ The simplest quadratic function is $y=x_{2},$ and the graph of any quadratic function is a parabola.

The inherent shape of parabolas gives rise to several characteristics that all quadratic functions have in common.
### Concept

### Direction

A parabola either opens **upward** or **downward**. This is called its direction.
### Concept

### Vertex

Because a parabola either opens upward or downward, there is always one point that is the absolute maximum or absolute minimum of the function. This point is called the vertex.
### Concept

### Axis of Symmetry

All parabolas are symmetric, meaning there exists a line that divides the graph into two mirror images. For quadratic functions, that line is always parallel to the $y$-axis, and is called the axis of symmetry.

### Concept

### Zeros

Depending on its rule, a parabola can intersect the $x$-axis at $0,$ $1,$ or $2$ points. Since the function's value at an $x$-intercept is always $0,$ these points are called zeros, or sometimes roots.
### Concept

### $y$-intercept

Because all graphs of quadratic functions extend infinitely to the left and right, they each have a $y$-intercept anywhere along the $y$-axis.

At the vertex, the function changes from increasing to decreasing, or vice versa.

The axis of symmetry always intersects the vertex of the parabola, and is written as a vertical line, where $h$ can be any real number.

$x=h$

When a quadratic function is written in standard form, it's possible to use $a,$ $b,$ and $c$ to determine characteristics of its graph.
$directiony-interceptaxis of symmetry :upward whena>0,:downward whena<0:(0,c):x=-2ab $
### Concept

### Direction

The direction of the graph is determined by the sign of $a.$ To understand why, consider the quadratic function
$y=ax_{2}.$
Since all squares are positive, $x_{2}$ will always be positive. When $a$ is positive, then $ax_{2}$ is also positive. Thus, when moving away from the origin in either direction, the graph extends upward. Similarly, when $a$ is negative, $ax_{2}$ will be negative. Thus, the graph will extend downward for all $x$-values.
### Concept

### $y$-intercept

The $y$-intercept of a quadratic function is given by $c,$ specifically at $(0,c).$ This is because substituting $x=0$ into standard form yields the following.
$yyyy =ax_{2}+bx+c=a⋅0_{2}+b⋅0+c=0+0+c=c $
### Concept

### Axis of Symmetry

The equation of the axis of symmetry can be found using the coefficients $a$ and $b.$ It is derived from the fact that the axis of symmetry divides the parabola in two mirror images. Two points with the same $y$-value are, thus, equidistant from the axis of symmetry. This gives rise to a quadratic equation where the solution is the axis of symmetry.

**Vertex form** is an algebraic format used to express quadratic function rules.

$y=a(x−h)_{2}+k$

In this form, $a$ gives the direction of the parabola. When $a>0,$ the parabola faces upward and when $a<0,$ it faces downward. The vertex of the parabola lies at $(h,k),$ and the axis of symmetry is $x=h.$ Consider the graph of $f(x)=-21 (x+4)_{2}+8.$

From the graph, we can connect the following characterisitcs to the function rule. $directionvertexaxis of symmetry :downward:(-4,8):x=-4 →a<0→h=-4,k=8→h=-4 $

Notice that although the factor in the function rule shows $(x+4)_{2},$ $h$ is actually equal to $-4.$ This coincides with a horizontal translation of a quadratic function.Quadratic function rules can be expressed in factored form, sometimes referred to as intercept form.

$y=a(x−s)(x−t)$

As is the case with standard form and vertex form, $a$ gives the direction of the parabola. When $a>0,$ the parabola faces upward, and when $a<0,$ it faces downward. Additionally, the zeros of the parabola lie at $(s,0)and(t,0).$ Because the points of a parabola with the same $y$-coordinate are equidistant from the axis of symmetry, the axis of symmetry lies halfway between the zeros. Consider the graph of $f(x)=(x+4)(x+2).$

From the graph, the following characteristics can be connected to the function rule.

$directionzerosaxis of symmetry :upward:(-4,0)&(-2,0):x=-3 →a>0→s=-4,t=-2→x=2-4+(-2) =-3 $For the following functions, determine the direction, the axis of symmetry, vertex, and zeros. $f(x)=(x +1)(x−5)g(x)=-2(x−3)_{2}+2h(x)=0.5x_{2}+5x+10.5 $

Show Solution

We'll focus on each function individually, starting with $f.$ ### Example

### Characteristics of $f(x)$

Notice that the function rule of $f$ is written in factored form since it's written as a product two binomials. $f(x)=(x+1)(x−5)$
We'll start by identifying $f$'s direction. It's determined by the sign of the coefficient in front of the factors. When a function doesn't have a number in front of the factors it can be interpreted as $1,$
$1⋅(x+1)(x−5).$
Since $1$ is positive, the parabola will face upward. Moving on, when a function is in factored form it's straightforward to find its zeros. They can be found by solving $f(x)=0.$ Using the Zero Product Property we find $x=-1$ and $x=5.$ Therefore, the zeros of the function are
$(-1,0)and(5,0).$
The axis of symmetry, axis of symmetry, can now be found using the zeros since it lies halfway between them. Add the $x$-coordinates and divide the sum by $2.$
Thus, the axis of symmetry is $x=2.$ Lastly, we can find the vertex by determining the function value of the axis of symmetry, $2.$
$f(2)=(2+1)(2−5)=-9$
The vertex is located at $(2,-9),$ and it is the absolute minimum of the function since $f$ opens upward. The characteristics of the function $f$ can be concluded as follows.
$directionaxis of symmetryvertexzeros :upward:x=2:absolute minimum at(2,-9):(-1,0)and(5,0) $
### Example

### Characteristics of $g(x)$

The function rule of $g$ is written in vertex form $a(x−h)_{2}+k.$ The direction is given by $a,$ the sign of the number in front of the binomial.
$g(x)=-2(x−3)_{2}+2$
Since $a$ is negative, the direction is downward. The vertex of the parabola is the point $(h,k).$
$g(x)=-2(x−3)_{2}+2$
Thus, the vertex of the function is $(3,2).$ Since the direction of the function is downward, the vertex is an absolute maximum. Because the axis of symmetry intersects the vertex, its equation is $x=3.$ What remains is to find the zeros of the function. This is done by solving the equation for $g(x)=0.$
Thus, the zeros of the function are $(2,0)$ and $(4,0).$ We have now found all of the characteristics of the function $g.$
$directionaxis of symmetryvertexzeros :downward:x=3:absolute maximum at(3,2):(2,0)and(4,0) $
### Example

### Characteristics of $h(x)$

The function rule of $h$ is expressed in standard form. Therefore, the values of $a,$ $b$ and $c$ can be used to determine its characteristics. The direction is given by the sign of $a,$ the coefficient in front of $x_{2}.$
$h(x)=0.5x_{2}+5x+10.5$
Since $a$ is positive, the direction is upward. For quadratic functions in standard form, the axis of symmetry can be found by
$x=-2ab .$
We already stated that $a=0.5,$ and $b$ is the coefficient of the $x$-term. Thus, $b=5.$ We can substitute $a$ and $b$ in the formula to find the axis of symmetry.
$x=-2⋅0.55 =-5$
Thus, the axis of symmetry is $x=-5.$ We can now find the vertex by calculating the function value $h(-5).$
The vertex of the function is $(-5,-2).$ Since the direction is upward, it's an absolute minimum. Finally, the zeros of $h$ can be found by solving the equation $h(x)=0.$ Since $h(x)$ is in standard form we'll use the quadratic formula.
Thus, the zeros are $(-7,0)$ and $(-3,0).$ We have now determined the characteristics for the function $h.$
$directionaxis of symmetryvertexzeros :downward:x=3:absolute maximum at(3,2):(2,0)and(4,0) $

$-2(x−3)_{2}+2=0$

$x_{1}=4x_{2}=2 $

$h(x)=0.5x_{2}+5x+10.5$

Substitute

$x=-5$

$h(-5)=0.5(-5)_{2}+5(-5)+10.5$

CalcPowProd

Calculate power and product

$h(-5)=0.5⋅25−25+10.5$

Multiply

Multiply

$h(-5)=12.5−25+10.5$

AddSubTerms

Add and subtract terms

$h(-5)=-2$

$0.5x_{2}+5x+10.5$

Solve using the quadratic formula

UseQuadForm

Use the Quadratic Formula: $a=0.5,b=5,c=10.5$

$x=2⋅0.5-5±5_{2}−4⋅0.5⋅10.5 $

CalcPowProd

Calculate power and product

$x=1-5±25−21 $

DivByOne

$1a =a$

$x=-5±25−21 $

SubTerm

Subtract term

$x=-5±4 $

CalcRoot

Calculate root

$x=-5±2$

StateSol

State solutions

$x_{1}=-3x_{2}=-7 $

A function's rate of change gives an indication of how its outputs ($y$) change with respect to its inputs ($x$).

$Rate of Change=ΔxΔy $

When the rate of change is constant, the function is linear. However, when the function is **not** linear, it's possible to determine an *average rate of change* over an arbitrary interval. This is an increase or decrease between the endpoints of the interval. If the $x$-coordinates of the endpoints are $x_{1}$ and $x_{2},$ and the function is $f,$ the average rate of change is defined as follows.

$Average Rate of Change=x_{2}−x_{1}f(x_{2})−f(x_{1}) $

The function $f(x)$ is quadratic.

Determine the average rate of change over the interval $[-3,2].$

Show Solution

To determine the average rate of change, we must know the $x$-values of the endpoints. Since the interval is $[-3,2],$ the $x$-values are $x=-3$ and $x=2.$ Let's mark these points on the graph to find their corresponding function values.

The $y$-values are $-2$ and $-7.$ Now, we can determine the average rate of change using the formula.$x_{2}−x_{1}f(x_{2})−f(x_{1}) $

SubstituteII

$x_{1}=-3$, $x_{2}=2$

$2−(-3)f(2)−f(-3) $

SubstituteII

$f(2)=-7$, $f(-3)=-2$

$2−(-3)-7−(-2) $

SubNeg

$a−(-b)=a+b$

$5-5 $

CalcQuot

Calculate quotient

$-1$

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