Represent the relationship shown in the table using words, an equation, and a graph. Then determine whether the relationship is a linear function.
Mountain Climbing
Number of Hours Climbing, x Elevation (ft), y
0 1103
1 1202
2 1301
3 1400
Represent the relationship using words. Choose the correct answer below.
A. Starting from a height of 1103 ft, you lose 99 ft each additional hour of climbing. B. Starting from a height of 99 ft, you gain 1103 ft each additional hour of climbing. OC. Starting from a height of 1103 ft, you gain 99 ft each additional hour of climbing. OD. Starting from a height of 99 ft, you lose 1103 ft each additional hour of climbing.
C. Starting from a height of 1103 ft, you gain 99 ft each additional hour of climbing.
Represent the relationship using an equation. y = ___ (Simplify your answer. Type an expression using x as the variable .)
The equation representing the relationship is y = 1103 + 99x, where x is the number of hours climbing and y is the elevation in feet.
Represent the relationship using a graph.
The relationship is a linear function because it can be represented by a straight line on a graph.
The relationship is a linear function.
Representing the relationship using words: Starting from a height of 1103 ft, you gain 99 ft each additional hour of climbing.
Representing the relationship using an equation: y = 1103 + 99x
To represent the relationship using an equation, we need to find a pattern or rule that connects the number of hours climbing (x) with the elevation (y). We observe that for each additional hour of climbing, the elevation increases by 99 ft. This means that the rate of change (or slope) is 99 ft per hour.
To find the equation, we use the point-slope form of a linear equation, which is:
y - y1 = m(x - x1)
where (x1, y1) is a point on the line, and m is the slope.
Using the first point (0, 1103), we have:
y - 1103 = 99(x - 0)
Simplifying this equation, we get:
y - 1103 = 99x
Therefore, the equation for the relationship is:
y = 99x + 1103
So, the equation is y = 99x + 1103.
Now let's determine whether this relationship is a linear function.
A linear function is a function where the dependent variable (in this case, elevation) varies linearly with the independent variable (in this case, number of hours climbing). In other words, the graph of a linear function is a straight line.
In this case, since the equation y = 99x + 1103 is in the form of a linear equation, it represents a linear function. Therefore, the relationship is a linear function.
Finally, let's represent the relationship using a graph.