Problem 57 - Understanding Elementary Algebra with Geometry

Question

Sketch the graph of \(h=-6 t+30\) using the horizontal axis for \(t\) values and the vertical axis for \(h\) values.

Step-by-Step Solution

Answer

Plot the y-intercept (0, 30), use the slope to find another point (1, 24), and draw the line through these points.

Step 1: Identify the slope and y-intercept

The given equation of the line is in slope-intercept form: \[ h = -6t + 30 \]Here, the slope (m) is -6 and the y-intercept (b) is 30.

Step 2: Plot the y-intercept

Start by plotting the y-intercept on the graph. The y-intercept occurs at \( t = 0 \). So, when \( t = 0 \), \( h = 30 \). Plot the point (0, 30) on the graph.

Step 3: Use the slope to find another point

The slope of -6 means that for every increase of 1 unit in \( t \), \( h \) decreases by 6 units. From the point (0, 30), move one unit to the right (increase \( t \) by 1), and then move 6 units down (decrease \( h \) by 6). This gives us the point (1, 24). Plot this point.

Step 4: Draw the line

Connect the points (0, 30) and (1, 24) with a straight line. This represents the graph of the equation \( h = -6t + 30 \). Extend the line in both directions.

Key Concepts

slope-intercept form

To understand how to graph linear functions like the one in the exercise, we first need to understand the slope-intercept form. The slope-intercept form is written as \[ y = mx + b \] where

• m is the slope of the line, which tells us how steep the line is.
• b is the y-intercept, which is the point where the line crosses the vertical (y) axis.

For example, in our exercise, the linear equation is written as \[ h = -6 t + 30 \] Here,

• The slope (m) is -6, which means the line slopes downwards. For every unit increase in t, h decreases by 6 units.
• The y-intercept (b) is 30. This is where the line crosses the y-axis.

graphing linear functions

Graphing linear functions involves several clear steps that make the process easier and error-free. Here's how you can graph any linear function:

• Identify the slope and y-intercept: As seen in our exercise, the given equation is already in slope-intercept form. Easily spot the slope and y-intercept.
• Plot the y-intercept: The y-intercept is where the line hits the y-axis. For our equation \( h = -6t + 30 \), the y-intercept is 30.
• Plot additional points using the slope: From the y-intercept, use the slope to find more points. Since our slope is -6, move one unit to the right and 6 units down to get another point on the line.
• Draw the line: Connect your points with a straight line and extend it in both directions.

For our specific example, we started at point (0, 30) and moved to point (1, 24) using the slope. Connecting these points gives us the graph of the equation \( h = -6 t + 30 \).

y-intercept

The y-intercept is a crucial concept when it comes to graphing linear equations. It's the point where the line crosses the y-axis, and it often provides our first starting point for graphing. Here's what you need to know about the y-intercept:

• Position on the Graph: For any linear function in the form \[ y = mx + b \] the y-intercept is the value of b. This is where our line intersects the y-axis when \[ x = 0 \] In our exercise, when t (or x) is 0, \[ h = 30 \] Therefore, the y-intercept is (0, 30).
• Starting Point for Graphing: Begin plotting your line by marking the y-intercept on the graph, then use the slope to plot more points.
• Vertical Crossing: The y-intercept tells us the value of the line at the exact moment it crosses the y-axis, providing a vital reference point for understanding where the line lies on the graph.

By understanding the y-intercept, we lay the foundation for accurately graphing any linear equation. The y-intercept directly influences the starting position of our graph, helping us visually understand how the line will extend across the graph.