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.
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(x−h)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.
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.
When a graph is shifted either horizontally or vertically, it is called a translation.
Horizontally: $g(x)=f(x-h)$g(x)=f(x−h) 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
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 subtract negative two from the $x$x-value inside parentheses: $f\left(x\right)=\left(x-\left(-2\right)\right)^2$f(x)=(x−(−2))2, which is the same as $f\left(x\right)=\left(x+2\right)^2$f(x)=(x+2)2. So shifts to the right are represented by subtracting a number inside the parentheses, while shifts to the left are represented by adding a number inside the parentheses.
What is the impact of the shift on the key features of the graph?
Now, let's explore the what happens when we shift the parent function up one unit. We will add one to the parent function, $x^2$x2: $f\left(x\right)=x^2+5$f(x)=x2+5. Vertical shifts up are represented by adding a number outside of the parentheses, while vertical shifts down are represented by subtracting a number outside of the parentheses.
What is the impact of the shift on the key features of the graph?
Dilation has the effect of stretching or compressing the parent function graph while the reflection flips the graph over the $x$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 sketching the graph of 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.
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 $x$x- and $y$y-intercepts are at the origin $\left(0,0\right)$(0,0).
The extrema is an absolute minimum of $0$0.
The domain of the function is all real numbers, $\left(-\infty,\infty\right)$(−∞,∞)
The range of the function is all real numbers greater than $0$0, $\left[0,\infty\right)$[0,∞)
The vertex, or turning point, is at the origin $\left(0,0\right)$(0,0). The axis of symmetry is the vertical line $x=0$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:
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.
Consider the graph of the function $y=f\left(x\right)$y=f(x) and answer the following questions.
What is the absolute minimum of the graph?
Hence determine the range of the function.
$y\ge\editable{}$y≥
Over what interval of the domain is the function increasing?
$x>\editable{}$x>
This is a graph of $y=x^2$y=x2.
How do we shift the graph of $y=x^2$y=x2 to get the graph of $y=\left(x-2\right)^2$y=(x−2)2?
Move the graph downwards by $2$2 units.
Move the graph to the right by $2$2 units.
Move the graph to the left by $2$2 units.
Move the graph upwards by $2$2 units.
Hence plot $y=\left(x-2\right)^2$y=(x−2)2 on the same graph as $y=x^2$y=x2.
Consider the equation $y=x^2+5$y=x2+5.
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{}$$)$), $G$G$($($3$3, $\editable{}$$)$)
Plot the points $C$C, $D$D and $E$E on the coordinate axes.
Plot the curve that results from the entire set of solutions for the equation being graphed.
Graph functions expressed algebraically and show key features of the graph both by hand and by using technology.
Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior.