Given a function and both a vertical and a horizontal shift, sketch the graph.
Identify the vertical and horizontal shifts from the formula.
The vertical shift results from a constant added to the output. Move the graph up for a positive constant and down for a negative constant.
The horizontal shift results from a constant added to the input. Move the graph left for a positive constant and right for a negative constant.
Apply the shifts to the graph in either order.
Graphing combined vertical and horizontal shifts
Given
sketch a graph of
The function
is our toolkit absolute value function. We know that this graph has a V shape, with the point at the origin. The graph of
has transformed
in two ways:
is a change on the inside of the function, giving a horizontal shift left by 1, and the subtraction by 3 in
is a change to the outside of the function, giving a vertical shift down by 3. The transformation of the graph is illustrated in
[link] .
Let us follow one point of the graph of
The point
is transformed first by shifting left 1 unit:
The point
is transformed next by shifting down 3 units:
Identifying combined vertical and horizontal shifts
Write a formula for the graph shown in
[link] , which is a transformation of the toolkit square root function.
The graph of the toolkit function starts at the origin, so this graph has been shifted 1 to the right and up 2. In function notation, we could write that as
Using the formula for the square root function, we can write
Graphing functions using reflections about the axes
Another transformation that can be applied to a function is a reflection over the
x - or
y -axis. A
vertical reflection reflects a graph vertically across the
x -axis, while a
horizontal reflection reflects a graph horizontally across the
y -axis. The reflections are shown in
[link] .
Vertical and horizontal reflections of a function.
Notice that the vertical reflection produces a new graph that is a mirror image of the base or original graph about the
x -axis. The horizontal reflection produces a new graph that is a mirror image of the base or original graph about the
y -axis.
Reflections
Given a function
a new function
is a
vertical reflection of the function
sometimes called a reflection about (or over, or through) the
x -axis.
Given a function
a new function
is a
horizontal reflection of the function
sometimes called a reflection about the
y -axis.
Given a function, reflect the graph both vertically and horizontally.
Multiply all outputs by –1 for a vertical reflection. The new graph is a reflection of the original graph about the
x -axis.
Multiply all inputs by –1 for a horizontal reflection. The new graph is a reflection of the original graph about the
y -axis.
Questions & Answers
Discuss the differences between taste and flavor, including how other sensory inputs contribute to our perception of flavor.
The lymphatic system plays several crucial roles in the human body, functioning as a key component of the immune system and contributing to the maintenance of fluid balance. Its main functions include:
1. Immune Response: The lymphatic system produces and transports lymphocytes, which are a type of
asegid
to transport fluids fats proteins and lymphocytes to the blood stream as lymph
Anatomy is the study of the structure of the body, while physiology is the study of the function of the body. Anatomy looks at the body's organs and systems, while physiology looks at how those organs and systems work together to keep the body functioning.
Enzymes are proteins that help speed up chemical reactions in our bodies. Enzymes are essential for digestion, liver function and much more. Too much or too little of a certain enzyme can cause health problems
Kamara
yes
Prince
how does the stomach protect itself from the damaging effects of HCl
the normal temperature is 37°c or 98.6 °Fahrenheit is important for maintaining the homeostasis in the body
the body regular this temperature through the process called thermoregulation which involves brain skin muscle and other organ working together to maintain stable internal temperature
