Question

Consider the linear function that is represented by the equation y = 4 x + 5 and the linear function that is represented by the table below.

x
y
2
16
4
26
6
36
8
46

Which statement is correct regarding their slopes and y-intercepts?
The function that is represented by the equation has a steeper slope and a greater y-intercept.
The function that is represented by the equation has a steeper slope, and the function that is represented by the table has a greater y-intercept.
The function that is represented by the table has a steeper slope, and the function that is represented by the equation has a greater y-intercept.
The function that is represented by the table has a steeper slope and a greater y-intercept.

Answer 1

To determine the slopes and y-intercepts of the two functions, we'll first analyze the given equation and the data in the table.

1. **For the function represented by the equation \(y = 4x + 5\)**:
- The slope \(m\) is \(4\).
- The y-intercept (the value of \(y\) when \(x=0\)) is \(5\).

2. **For the function represented by the table**:
- We can identify the slope by using the change in \(y\) divided by the change in \(x\).
- From the table:
- For \(x = 2\), \(y = 16\)
- For \(x = 4\), \(y = 26\)
- The change from \(x = 2\) to \(x = 4\) is \(26 - 16 = 10\) and the change in \(x\) is \(4 - 2 = 2\).
- So, slope \(m = \frac{10}{2} = 5\).

- The y-intercepts can be calculated by observing the values in the table.
- At \(x = 2\), \(y = 16\).
- At \(x = 0\) (if we backtrack), if we decrease \(x\) to 0, we can see from the equation of the line formed by the points in the table:
- The slope from the previous calculation is \(5\), and so the equation can be expressed in slope-intercept form as \(y = 5x + b\).
- Using point \((2, 16)\):
\[
16 = 5(2) + b \implies 16 = 10 + b \implies b = 6
\]
- Thus, the y-intercept is \(6\).

3. **Comparing the two functions**:
- The function from the equation has a slope of \(4\) and a y-intercept of \(5\).
- The function from the table has a slope of \(5\) and a y-intercept of \(6\).

### Conclusion:
- The slope of the function from the table (\(5\)) is steeper than the slope of the function represented by the equation (\(4\)).
- The y-intercept of the function from the table (\(6\)) is greater than the y-intercept of the function represented by the equation (\(5\)).

### Therefore, the correct statement is:
**The function that is represented by the table has a steeper slope and a greater y-intercept.**