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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