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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
A reflection of a function is a transformation that flips a graph over a line called the line of reflection. We can make a reflection in the y-axis by changing the sign of every input of the given function rule. This means that we need to find an expression for f( - x). g(x)=f( -x) ⇕ g(x)=- 2( -x)^2+12( -x)-16 Let's simplify the obtained equation!
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
We can make a reflection in the x-axis by changing the sign of every output of the given function rule. This means that we need to find an expression for - f(x). h(x)= -f(x) ⇕ h(x)= - (- 2x^2+12x-16) Let's simplify the obtained equation by distributing - 1.
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
A function graph is vertically stretched by multiplying every output by a positive constant greater than 1. Therefore, if we want to vertically stretch the graph of the given function by a factor of 2, we multiply the function rule by 2. This means that we need to find an expression for 2f(x). g(x)=2f(x) ⇕ g(x)=2(- 2x^2+12x-16) We simplify the obtained equation by distributing 2.
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
A function graph is vertically shrunk by multiplying every output by a positive constant less than 1. Therefore, if we want to vertically shrink the graph of the given function by a factor of 12, we multiply the function rule by 12. This means that we need to find an expression for 12f(x). h(x)=1/2f(x) ⇕ h(x)=1/2(- 2x^2+12x-16) We simplify the obtained equation by distributing 12.
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
A function graph is horizontally stretched by multiplying every input value by a positive constant less than 1. Therefore, if we want to horizontally stretch the graph of the given function by a factor of 12, we multiply the variable x by 12. This means that we need to find an expression for f( 12x). g(x)=f(1/2x) ⇕ g(x)=2(1/2x)^2-2 Let's simplify the obtained equation by applying properties of exponents.
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
A function graph is horizontally shrunk by multiplying every input value by a positive constant greater than 1. Therefore, if we want to horizontally shrink the graph of the given function by a factor of 2, we multiply the variable x by 2. This means that we need to find an expression for f(2x). h(x)=f(2x) ⇕ h(x)=2(2x)^2-2 Let's simplify the obtained equation!
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
A translation of a function is a transformation that shifts a graph vertically or horizontally. We do a vertical translation by adding some number to every output value of a function rule. Therefore, if we want to translate the graph of f(x) 5 units up, we need to find an expression for f(x)+5. g(x)=f(x)+5 ⇕ g(x)=- 2x^2+12x-16+5 We can simplify the obtained equation by combining like terms.
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
Previously, we said that a translation of a function is a transformation that shifts a graph vertically or horizontally. We perform a horizontal translation by subtracting some number to every input value of a function rule. Therefore, if we want to translate the graph of f(x) 4 units to the right, we need to find an expression for f(x-4). h(x)=f(x-4) ⇕ h(x)=- 2(x-4)^2+12(x-4)-16 Let's simplify the obtained equation!
We can verify the obtained result by graphing both quadratic functions on the same coordinate plane.
We want to describe how the graph of the given quadratic function is related to the graph of its parent function f(x)=x^2. To describe their relation, we will go through the process of transforming the parent function's graph of f(x)=x^2 into the graph of the given function. h(x)=1/2(x-2)^2 In order to do this, we need to consider two possible transformations.
Let's consider them one at a time.
We have a vertical stretch when the function rule is multiplied by a positive number greater than one. If the output is multiplied by a positive number less than one, a vertical shrink takes place.
If the function rule is multiplied by a negative number, then a reflection takes place. In our case, the output of the function is multiplied by 12. Therefore, the graph of the parent function is vertically shrunk by a factor of 12.
If an addition or subtraction is applied only to the x-variable, the graph is horizontally translated. In cases of addition, the graph is translated to the left. In cases of subtraction, it is moved to the right. In the given equation, 2 is being subtracted from x, so the previous graph is translated 2 units to the right.
Let's now graph the given function and the parent function f(x)=x^2 on the same coordinate plane.
Finally, let's summarize how the given function relates to the parent function f(x)=x^2.
We want to describe how the graph of the given quadratic function is related to the graph of its parent function f(x)=x^2. We will do this by transforming the graph of f(x)=x^2 into the graph of the given function. h(x)=- x^2-2 To make this transformation, we need to consider two possible types.
Let's consider them one at a time.
Keep in mind that when a function rule is multiplied by - 1, the graph is reflected in the x-axis. Our given equation includes - x^2, indicating a reflection about the x-axis. Let's take a look at its graph.
Note how each x-coordinate stays the same, while on the other hand, each y-coordinate changes its sign.
If an addition or subtraction is applied to the whole function, the graph is vertically translated. In cases of addition, the graph is translated up. In cases of subtraction, it is moved downwards. In the given equation, 2 is subtracted from the whole function, which means that our graph is translated 2 units down.
Let's now graph the given function and the parent function f(x)=x^2 on the same coordinate plane.
Finally, let's summarize how the graph of the given function relates to the parent function f(x)=x^2.