Problem 1
Question
Suppose we want to graph an equation in \(x\) and \(y .\) (a) If we replace \(x\) by \(x-3,\) the graph of the equation is shifted to the _____ by 3 units. If we replace \(x\) by \(x+3,\) the graph of the equation is shifted to the ____ by 3 units. (b) If we replace \(y\) by \(y-1,\) the graph of the equation is shifted _____ by 1 unit. If we replace \(y\) by \(y+1,\) the graph of the equation is shifted _____ by 1 unit.
Step-by-Step Solution
Verified Answer
(a) right, left; (b) up, down.
1Step 1: Identifying Horizontal Shifts
To understand horizontal shifts in the graph of an equation, consider replacing the variable - When we replace \(x\) with \(x-3\), the value of \(x\) increases by 3 (to become equal to the former value of \(x\)). This shifts the graph to the right by 3 units.- Conversely, replacing \(x\) with \(x+3\) decreases the effective value of \(x\) by 3, shifting the graph to the left by 3 units.
2Step 2: Identifying Vertical Shifts
Vertical shifts occur when altering the \(y\)-variable in the equation as follows:- Replacing \(y\) with \(y-1\) implies that each value of \(y\) increases by 1, moving the graph upward by 1 unit.- Replacing \(y\) with \(y+1\) means the effective \(y\) becomes the original value decremented by 1, thus shifting the graph downward by 1 unit.
Key Concepts
Horizontal ShiftsVertical ShiftsCoordinate System
Horizontal Shifts
Horizontal shifts involve moving the graph of a function along the x-axis. This is achieved by changing the variable in the equation by adding or subtracting a constant.
When we replace the variable \( x \) with \( x-3 \), it essentially means that every \( x \) value on the graph is increased by 3 units. As a result, the entire graph shifts to the right by 3 units.
On the other hand, replacing \( x \) with \( x+3 \) decreases the effective \( x \) values by 3 units, shifting the graph to the left by 3 units.
This concept is useful to understand because small shifts in the graph can significantly change the visual output without altering the core property of the function.
When we replace the variable \( x \) with \( x-3 \), it essentially means that every \( x \) value on the graph is increased by 3 units. As a result, the entire graph shifts to the right by 3 units.
On the other hand, replacing \( x \) with \( x+3 \) decreases the effective \( x \) values by 3 units, shifting the graph to the left by 3 units.
This concept is useful to understand because small shifts in the graph can significantly change the visual output without altering the core property of the function.
Vertical Shifts
Vertical shifts affect how a graphed function moves along the y-axis. By adjusting the \( y \)-values of an equation, we can move the entire graph up or down.
When we substitute \( y \) with \( y-1 \), it indicates that each \( y \) value on the graph effectively becomes 1 unit larger. This causes the graph to shift upward by 1 unit.
Conversely, if we replace \( y \) with \( y+1 \), it implies each \( y \) decreases effectively by 1 unit, thus shifting the graph downward by 1 unit.
Vertical shifts ensure that while the graph changes its position, the shape of the graph remains unchanged. This operation is vital for moving a graph without altering its basic characteristics.
When we substitute \( y \) with \( y-1 \), it indicates that each \( y \) value on the graph effectively becomes 1 unit larger. This causes the graph to shift upward by 1 unit.
Conversely, if we replace \( y \) with \( y+1 \), it implies each \( y \) decreases effectively by 1 unit, thus shifting the graph downward by 1 unit.
Vertical shifts ensure that while the graph changes its position, the shape of the graph remains unchanged. This operation is vital for moving a graph without altering its basic characteristics.
Coordinate System
Understanding the coordinate system is fundamental to graph transformations. The coordinate system comprises two perpendicular lines: the horizontal x-axis and the vertical y-axis. These axes help to display and analyze the position of points in a two-dimensional space.
- **Origin**: The point where both axes intersect, denoted as (0,0). - **Quadrants**: The coordinate plane is divided into four quadrants, each representing a different combination of positive and negative values of x and y.
When performing shifts, it is crucial to understand how the coordinates of each point on a graph are affected. For horizontal shifts, only the x-coordinate changes, while for vertical shifts, only the y-coordinate changes.
This knowledge helps in interpreting and predicting graph movements efficiently, maintaining clarity and accuracy in visualizing mathematical functions.
- **Origin**: The point where both axes intersect, denoted as (0,0). - **Quadrants**: The coordinate plane is divided into four quadrants, each representing a different combination of positive and negative values of x and y.
When performing shifts, it is crucial to understand how the coordinates of each point on a graph are affected. For horizontal shifts, only the x-coordinate changes, while for vertical shifts, only the y-coordinate changes.
This knowledge helps in interpreting and predicting graph movements efficiently, maintaining clarity and accuracy in visualizing mathematical functions.
Other exercises in this chapter
Problem 1
An ellipse is the set of all points in the plane for which the ________ of the distances from two fixed points \(F_{1}\) and \(F_{2}\) is constant. The points \
View solution Problem 1
A parabola is the set of all points in the plane that are equidistant from a fixed point called the _____ and a fixed line called the _____ of the parabola.
View solution Problem 2
The graphs of \(x^{2}=12 y\) and \((x-3)^{2}=12(y-1)\) are given. Label the focus, directrix, and vertex on each parabola. (GRAPH NOT COPY)
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