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: Understand horizontal shifts
When a function is replaced by \( f(x) \) with \( f(x-3) \), the graph shifts to the right by 3 units. Conversely, replacing \( f(x) \) with \( f(x+3) \) shifts the graph to the left by 3 units.
2Step 2: Identify horizontal shifts for the given exercise
Based on Step 1, replacing \( x \) with \( x-3 \) shifts the graph to the **right** by 3 units, and replacing \( x \) with \( x+3 \) shifts the graph to the **left** by 3 units.
3Step 3: Understand vertical shifts
When a function is replaced by \( f(y) \) with \( f(y-1) \), it shifts the graph up by 1 unit. This is because we need to _decrease_ \( y \) to achieve the original value in the function, resulting in an upward shift. Conversely, replacing \( f(y) \) with \( f(y+1) \) shifts the graph down by 1 unit.
4Step 4: Identify vertical shifts for the given exercise
Using the logic from Step 3, replacing \( y \) with \( y-1 \) shifts the graph **up** by 1 unit, and replacing \( y \) with \( y+1 \) shifts the graph **down** by 1 unit.
Key Concepts
Horizontal ShiftsVertical ShiftsCoordinate Plane
Horizontal Shifts
When we talk about horizontal shifts in graph transformations, we're moving the graph along the x-axis. This is done by changing the value of x in the equation of the function.
For example, if you have a function \( f(x) \), and you change it to \( f(x-3) \), the whole graph shifts to the right by 3 units. Why right? Because you're essentially "delaying" the effect of x by 3, meaning you need a larger x to reach the same output as before.
On the other hand, if you replace \( x \) by \( x+3 \), the graph shifts to the left by 3 units. Think of it like you're preemptively adding 3 units to x, so it requires less x to achieve the previous outputs.
For example, if you have a function \( f(x) \), and you change it to \( f(x-3) \), the whole graph shifts to the right by 3 units. Why right? Because you're essentially "delaying" the effect of x by 3, meaning you need a larger x to reach the same output as before.
On the other hand, if you replace \( x \) by \( x+3 \), the graph shifts to the left by 3 units. Think of it like you're preemptively adding 3 units to x, so it requires less x to achieve the previous outputs.
- Replacing x with \( x-3 \): shift **right** by 3 units
- Replacing x with \( x+3 \): shift **left** by 3 units
Vertical Shifts
Vertical shifts occur when the graph moves up or down along the y-axis. This happens when we change the value of y in the equation. It might seem counterintuitive initially, but it's pretty simple.
Let's say you change the function \( f(y) \) to \( f(y-1) \). This transformation shifts the graph *up* by 1 unit. It's like you have to increase y to get back to your original function value.
If instead, you change the equation to \( f(y+1) \), the graph shifts *downward* by 1 unit. Here, because you're "prematurely" increasing y, you effectively move everything lower to maintain the same functional values.
Let's say you change the function \( f(y) \) to \( f(y-1) \). This transformation shifts the graph *up* by 1 unit. It's like you have to increase y to get back to your original function value.
If instead, you change the equation to \( f(y+1) \), the graph shifts *downward* by 1 unit. Here, because you're "prematurely" increasing y, you effectively move everything lower to maintain the same functional values.
- Replacing y with \( y-1 \): shift **up** by 1 unit
- Replacing y with \( y+1 \): shift **down** by 1 unit
Coordinate Plane
Understanding the coordinate plane is crucial for graph transformations. It consists of two perpendicular axes: the x-axis (horizontal) and the y-axis (vertical). Together, they form a grid where you can plot points representing equations.
Each point on this plane is given by a pair of coordinates \((x, y)\). These coordinates help locate the point precisely on the plane. Horizontal and vertical shifts manipulate these coordinates by adding or subtracting from x or y.
Each point on this plane is given by a pair of coordinates \((x, y)\). These coordinates help locate the point precisely on the plane. Horizontal and vertical shifts manipulate these coordinates by adding or subtracting from x or y.
- Horizontal shifts affect the x-coordinates.
- Vertical shifts affect the y-coordinates.
Other exercises in this chapter
Problem 1
A hyperbola 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
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 p
View solution Problem 2
The graph of the equation \(\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1\) with \(a>0, b>0\) is a hyperbola with vertices (________ , ________) and (_______ , ____
View solution