Problem 31 - Thomas Calculus

Question

Exercises \(25-34\) tell how many units and in what directions the graphs of the given equations are shifted. Give an equation for the shifted graph. Then sketch the original and shifted graphs together, labeling each graph with its equation. $$ y=2 x-7 \quad \text { Up } 7 $$

Step-by-Step Solution

Answer

The graph is shifted 7 units up, resulting in the equation \(y = 2x\).

Step 1: Identify Shift

The problem states that the graph needs to be shifted 'Up 7.' This means we need to add 7 to the y-coordinate in the equation.

Step 2: Apply Shift to Equation

Start with the original equation: \[ y = 2x - 7 \]Since we are shifting the graph 7 units up, we add 7 to the y-coordinate: \[ y = 2x - 7 + 7 \]Running the calculations, we find:\[ y = 2x \]This is the equation for the shifted graph.

Step 3: Sketch Graphs

To sketch the graphs, first draw the line for the original equation, \( y = 2x - 7 \), which is a straight line with a slope of 2 and y-intercept at -7. Then, draw the shifted graph using the equation \( y = 2x \). This line also has a slope of 2 but its y-intercept is at 0. The shifted graph is parallel to the original graph and shifted up 7 units.

Key Concepts

Linear Equations

Linear equations are mathematical expressions that create a straight line when graphed on a coordinate plane. These equations are written in the form of \( y = mx + b \), where \( m \) represents the slope, \( x \) is the variable, and \( b \) is the y-intercept. A linear equation describes a constant rate of change, giving it a uniform slant when graphed. This uniformity makes them relatively straightforward and predictable when transforming or manipulating their graphs.Understanding linear equations is essential because they form the basis for more complex equations and represent many real-world relationships. By adjusting the slope or y-intercept, you can easily model different scenarios and see how varying these parameters affects the graph.

Graph Shifting

Graph shifting involves moving a graph vertically or horizontally on a coordinate plane without changing its overall shape. In the context of linear equations, graph shifting is typically achieved by altering the y-intercept value.When you shift a graph upwards, you add a constant to the entire equation. For instance, shifting the equation \( y = 2x - 7 \) upwards by 7 units results in the new equation \( y = 2x \). This process effectively moves every point on the line 7 units up, keeping the slope constant, and results in a parallel line.

• Vertical shifts involve adding or subtracting from the entire equation.
• Horizontal shifts would involve changing the x-value, though not common for vertical transformations.

Shifting graphs allows you to better understand transformations and how equations translate into geometric space.

Slope and Y-intercept

The slope \( m \) and y-intercept \( b \) are fundamental components of the linear equation \( y = mx + b \). Understanding these concepts is key to mastering graph transformations.The slope describes the line's steepness and direction. A positive slope, like 2 in the equation \( y = 2x - 7 \), indicates that as you move from left to right, the line rises. The number "2" here signifies that for every 1 unit increase in \( x \), \( y \) increases by 2 units. On the other hand, the y-intercept is where the line crosses the y-axis. In the original equation, \( y = 2x - 7 \), the y-intercept is -7, meaning the line begins at the point (0, -7) when graphed. After shifting "Up 7," the y-intercept becomes 0.

• The slope determines the angle and direction of the line.
• The y-intercept sets the starting point of the line on the y-axis.

By manipulating these values, you can see how they affect the position and orientation of the line on a graph. Being comfortable with slopes and intercepts is crucial for graph transformations and understanding linear behavior.