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| 14 Theory slides |
| 10 Exercises - Grade E - A |
| Each lesson is meant to take 1-2 classroom sessions |
Here are a few recommended readings before getting started with this lesson.
The graph of the quadratic function f(x)=-x2+2x+1 is drawn on the coordinate plane. By applying transformations to the parabola that corresponds to f, draw the graph of g(x)=-x2+4x.
Kriz is given an extra credit math assignment about quadratic functions and parabolas. The assignment consists of four tasks. For the first task, Kriz is given the graph of the function y=21(x+2)2+1.
Help Kriz with their extra credit assignment.
Equation: y=-21(x+2)2−1
Equation: y=21(x−2)2+1
In the coordinate plane, the parabola that corresponds to the quadratic function y=af(bx) can be seen. Observe how the graph is vertically and horizontally stretched and shrunk by changing the values of a and b.
Vertical | Horizontal | |
---|---|---|
Stretch | af(x), with a>1 | f(ax), with 0<a<1 |
Shrink | af(x), with 0<a<1 | f(ax), with a>1 |
Kriz's second task is about horizontal and vertical stretches and shrinks of parabolas.
They are given the quadratic function y=x2−3 and want to write the function rules of two related functions.
The graph of the quadratic function y=x2 is shown in the coordinate plane. The graph of a horizontal or vertical stretch or shrink is also shown.
Kriz's assignment is getting more interesting, as the third task is about combining reflections with vertical and horizontal stretches and shrinks.
This time, they are given the quadratic function y=(x−1)2 and want to write the function rules of two other functions.
(-a)2=a2
Distribute -1
(a+b)2=a2+2ab+b2
1a=1
Identity Property of Multiplication
Distribute 3
(a−b)2=a2−2ab+b2
(ab)m=ambm
1a=1
Multiply
Distribute -1
In the coordinate plane, the graph of the quadratic function y=(x−h)2+k can be seen. Observe how the graph is horizontally and vertically translated by changing the values of h and k.
Translation | |
---|---|
Vertical | Horizontal |
Upwards f(x)+k, with k>0 |
To the Right f(x−h), with h>0 |
Downwards f(x)+k, with k<0 |
To the Left f(x−h), with h<0 |
To finally finish the assignment and get the extra credit they need, Kriz has to finish the fourth task of the math assignment. This time, the graph of the quadratic parent function y=x2 is given.
By translating this quadratic function, Kriz wants to draw the graphs and write the equations of the following functions.
Graph:
Graph:
Graph:
(a−b)2=a2−2ab+b2
Commutative Property of Multiplication
Calculate power
Multiply
(a+b)2=a2+2ab+b2
Commutative Property of Multiplication
Calculate power
Multiply
Subtract term
The graph of the quadratic function y=x2 and a vertical or horizontal translation are shown in the coordinate plane.
With the topics learned in this lesson, the challenge presented at the beginning can be solved. The parabola that corresponds to the quadratic function f(x)=-x2+2x+1 is given.
Rewrite the function whose graph is to be drawn to clearly identify the transformations.
Write as a difference
Factor out 3
(-a)2=a2
a+(-b)=a−b
Commutative Property of Multiplication
Two quadratic functions are graphed on the same coordinate plane.
We can see in the given diagram that the graph of h(x) is a translation 5 units to the left of the graph of f(x).
To translate the graph of a function 5 units to the left, we add 5 to the function's input. This means that we need to calculate an expression for f(x+5). h(x)=f(x+5) ⇕ h(x)=(x+5)^2+(x+5)-1 Let's simplify the right-hand side of this equation.
To check our answer, we can draw the graph of h(x)=x^2+11x+29 and verify that this parabola matches the given graph. To do so, we will first make a table of values.
x | x^2+11x+29 | h(x) |
---|---|---|
- 7 | ( - 7)^2+11( - 7)+29 | 1 |
- 6 | ( - 6)^2+11( - 6)+29 | - 1 |
- 5 | ( - 5)^2+11( - 5)+29 | - 1 |
- 4 | ( - 4)^2+11( - 4)+29 | 1 |
By using this table, we obtained the points (- 7,1), (- 6,- 1), (- 5,- 1), and (- 4,1). Let's plot and connect them with a smooth curve. We will also include the graph of f(x)=x^2+x-1 on the same coordinate plane to see their relationship.
Two quadratic functions are graphed on the same coordinate plane.
We can see in the given diagram that the graph of h(x) is wider than the graph of f(x). We can also see that both parabolas have the same vertex. These things suggest that the graph of h(x) is a horizontal stretch of the graph of f(x).
What is more, by counting the units, we can see that the graph of h(x) is four times wider than the graph of f(x). Therefore, the graph of h(x) is a horizontal stretch of the graph of f(x) by a factor of 14. To horizontally stretch the graph of a function by a factor of 14, we multiply the function's input by 14. This means that we need to find an expression for f( 14x). h(x)=f( 14x) ⇕ h(x)=(1/4x)^2-1 Let's simplify the right-hand side of the above equation by using properties of exponents.
Two quadratic functions are graphed on the same coordinate plane.
We can see in the given diagram that the x-axis acts like a mirror. Therefore, the graph of h(x) is a reflection of the graph of f(x) in the x-axis.
To reflect the graph of a function in the x-axis, we multiply the function rule by - 1. This means that we need to find an expression for - f(x). h(x)=- f(x) ⇕ h(x)=- (x^2-4x+1) We can simplify the above equation by distributing - 1 on the right-hand side.
To check our answer, we can draw the graph of h(x)=- x^2+4x-1 and verify that this parabola matches the given graph. To do so, we will first make a table of values.
x | - x^2+4x-1 | h(x) |
---|---|---|
0 | - 0^2+4( 0)-1 | - 1 |
1 | - 1^2+4( 1)-1 | 2 |
2 | - 2^2+4( 2)-1 | 3 |
3 | - 3^2+4( 3)-1 | 2 |
4 | - 4^2+4( 4)-1 | - 1 |
By using this table, we obtained the points (0,- 1), (1,2), (2,3), (3,2), and (4,- 1). Let's plot and connect them with a smooth curve. We will also include the graph of f(x)=x^2-4x+1 on the same coordinate plane to see their relationship.