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2.04 Review: Graphs and characteristics of quadratic functions

Lesson

Identifying key features

In Algebra I, we explored a few key features of quadratic functions. In this lesson, we are going to continue to distribute our understanding of quadratic functions.

Remember!

A quadratic function is a nonlinear function that can be written in general form:

$y=ax^2+bx+c$y=ax2+bx+c, where $a$a $\ne$ $0$0 and vertex occurs at $x=\frac{-b}{2a}$x=b2a

and can be written in vertex form (or standard form):

$y=a\left(x-h\right)^2+k$y=a(xh)2+k, where $a$a $\ne$  $0$0 and vertex is $\left(h,k\right)$(h,k)

The U-shaped graph of a quadratic function is called a parabola.

The quadratic function produces a continuous function.

 

Transformational graphing

A parent function is the most basic function of a family of functions. It preserves the shape of the entire family. The graph below models the function $f\left(x\right)=x^2$f(x)=x2, which is most basic quadratic function. 

The $x$x - and $y$y - intercepts are at the origin $(0,0)$(0,0).

The extrema is an absolute minimum at $0$0.

The domain of the function is all real numbers, $(\infty,\infty)$(,)

The range of the function is all real numbers greater than $0$0$[0,\infty)$[0,)

The vertex (turning point) is at the origin $(0,0)$(0,0) and the line of symmetry is $x=0$x=0

 

Translation

When a graph is shifted either horizontally or vertically, it is called a translation. 

Horizontally: $g(x)=f(x-h)$g(x)=f(xh) is $f\left(x\right)$f(x) translated $h$h units right when $h>0$h>0 or $h$h units left when $h<0$h<0

 

Vertically: $g\left(x\right)=f\left(x\right)+k$g(x)=f(x)+k is $f\left(x\right)$f(x) translated $k$k units up when $k>0$k>0 or $k$k units down when $k<0$k<0

 

Exploration

Let's say we want to move our parent graph of $f\left(x\right)=x^2$f(x)=x2 to the left two units. To do this, we have to subtract a negative two from the x value inside parentheses like so: $f\left(x\right)=\left(x-\left(-2\right)\right)^2$f(x)=(x(2))2, which is $f\left(x\right)=\left(x+2\right)^2$f(x)=(x+2)2 . Any shifts to the right will be completed through subtracting number inside the parentheses, while any shifts to the left will done be by adding a number inside the parentheses.

What is the impact of the shift on the key features of the graph?

The $x$x - and $y$y - intercepts are at the origin $\left(0,0\right)$(0,0).

The extrema is an absolute minimum at $0$0.

The domain of the function is all real numbers,$(\infty,\infty)$(,)

The range of the function is all real numbers greater than $0$0$[0,\infty)$[0,)

The vertex (turning point) is at the origin $\left(-2,0\right)$(2,0) and the line of symmetry is $x=-2$x=2.

 

Now, let's explore the what happens when we shift the parent function up one unit. We will need to add one to the parent function, $x^2$x2$f\left(x\right)=x^2+5$f(x)=x2+5 . Any vertical shifts up will be done by adding a number outside of the parentheses, while any vertical shifts down will come from subtracting a number outside of the parentheses.

What is the impact of the shift on the key features of the graph?

The $x$x- and $y$y- intercepts are at the origin $\text{(0, 0)}$(0, 0).

The extrema is an absolute minimum at $1$1.

The domain of the function is all real numbers, $(\infty,\infty)$(,)

The range of the function is all real numbers greater than or equal to $1$1 , $[1,\infty)$[1,)

The vertex (turning point) is at the origin $\text{(0, 1)}$(0, 1)and the line of symmetry is $x=0$x=0

 

Dilation and reflection

Dilation has the effect of stretching or compressing the parent function graph while the reflection flips the graph along the x-axis.

Dilation: $g\left(x\right)=af\left(x\right)$g(x)=af(x) if $a>1$a>1 stretches away from $x$x-axis; if $00<a<1 compresses towards $x$x-axis

 

Reflection: $g\left(x\right)=-f\left(x\right)$g(x)=f(x) reflected over the $x$x-axis

Note: When graphing a reflection, the ordered pair changes from $\left(x,y\right)$(x,y)  $\longrightarrow$ $\left(x,-y\right)$(x,y) for a reflection in the $x$x-axis.

 

Exploration

When given the parent function, $y=x^2$y=x2, is multiplied by a real number greater than $0$0, the parabola changes its shape. It remains a U-shape; however the U-shape becomes either compressed or stretched. The graph below shows the parent function, $y=x^2$y=x2, and two other functions. 

What do we notice about the change in shape of the parabola:

  • The parabola $y=3x^2$y=3x2 is steeper (compressed) than its parent function.
  • The $y$y-value of each point on $y=3x^2$y=3x2 parabola is three times the $y$y-value of the point on its parent function with the same $x$x-value. Hence, the graph has been compressed by a factor of $3$3
  • The parabola $y=0.5x^2$y=0.5x2 is not as steep as its parent function.
  • The $y$y-value of each point on $y=0.5x^2$y=0.5x2 parabola is half the y-value of the point on its parent function with the same $x$x-value. Hence, the graph has been stretched by a factor of $0.5$0.5
  • The following is unchanged from the parent function:
    • The $x$x- and $y$y- intercepts are at the origin $\left(0,0\right)$(0,0).

    • The extrema is an absolute minimum at $0$0.

    • The domain of the function is all real numbers, $(\infty,\infty)$(,)

    • The range of the function is all real numbers greater than $0$0, $[0,\infty)$[0,)

    • The vertex (turning point) is at the origin $\left(0,0\right)$(0,0) and the line of symmetry is $x=0$x=0

 

Now, what happens when we multiply the parent parent function by a negative $1$1

What do we notice about the change in shape of the parabola:

  • The parabola $y=-x^2$y=x2 maintains the same shape as its parent function. 
  • The ordered pairs on $y=3x^2$y=3x2 parabola are the opposite of those on its parent function: $\left(x,y\right)$(x,y) $\longrightarrow$$\left(-x,-y\right)$(x,y). Hence, the graph has been flipped across the x-axis
  • The parabola does not have an absolute minimum. It does have an absolute maximum at $0$0
  • The range of the graph $y=-x^2$y=x2 is all real numbers less than or equal to $0$0$(-\infty,0]$(,0]
  • The following is unchanged from the parent function:
    • The $x$x- and $y$y- intercepts are at the origin $\left(0,0\right)$(0,0).
    • The domain of the function is all real numbers, $(\infty,\infty)$(,)
    • The vertex (turning point) is at the origin $\left(0,0\right)$(0,0) and the line of symmetry is $x=0$x=0.

 

Increasing and decreasing intervals

A function can be described as increasing, decreasing, or constant over a specified interval or the entire domain. Since a parabola is U-shaped, the graph has specific intervals where it increases and decreases. Because the vertex is the greatest or least point on a parabola, its y-coordinate is the maximum value or minimum value of the function. The vertex of a parabola lies on its line of symmetry. So, the graph of the function is increasing on one side of the line of symmetry and decreasing on the other side.

Notice, that the intervals do not include $0$0. The vertex is the point where the graph changes direction; therefore this point cannot be increasing or decreasing.  

 

Practice questions

question 1

Consider the graph of the function $y=f\left(x\right)$y=f(x) and answer the following questions.

Loading Graph...

  1. What is the absolute minimum of the graph?

  2. Hence determine the range of the function.

    $y\ge\editable{}$y

  3. Over what interval of the domain is the function increasing?

    $x>\editable{}$x>

question 2

This is a graph of $y=x^2$y=x2.

Loading Graph...

  1. How do we shift the graph of $y=x^2$y=x2 to get the graph of $y=\left(x-2\right)^2$y=(x2)2?

    Move the graph downwards by $2$2 units.

    A

    Move the graph to the right by $2$2 units.

    B

    Move the graph to the left by $2$2 units.

    C

    Move the graph upwards by $2$2 units.

    D

    Move the graph downwards by $2$2 units.

    A

    Move the graph to the right by $2$2 units.

    B

    Move the graph to the left by $2$2 units.

    C

    Move the graph upwards by $2$2 units.

    D
  2. Hence plot $y=\left(x-2\right)^2$y=(x2)2 on the same graph as $y=x^2$y=x2.

    Loading Graph...

question 3

Consider the equation $y=x^2+5$y=x2+5.

  1. Complete the set of solutions for the given equation.

    $A$A$($($-3$3, $\editable{}$$)$), $B$B$($($-2$2, $\editable{}$$)$), $C$C$($($-1$1, $\editable{}$$)$), $D$D$($($\editable{}$, $5$5$)$), $E$E$($($1$1, $\editable{}$$)$), $F$F$($($2$2, $\editable{}$$)$), $G$G$($($3$3, $\editable{}$$)$)

  2. Plot the points $C$C, $D$D and $E$E on the coordinate axes.

    Loading Graph...

  3. Plot the curve that results from the entire set of solutions for the equation being graphed.

    Loading Graph...

Outcomes

A2.APR.A.5

Identify zeros of polynomials when suitable factorizations are available, and use the zeros to sketch the function defined by the polynomial.

A2.IF.A.1

Identify and interpret key characteristics of functions represented graphically, with tables and with algebraic symbolism to solve problems.

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