To determine which of the given equations represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex], we need to look for an equation with the same slope and [tex]\( y \)[/tex]-intercept as the given line equation. The given equation [tex]\( f(x) = 4x + 3 \)[/tex] is in slope-intercept form [tex]\( y = mx + b \)[/tex], where the slope [tex]\( m \)[/tex] is 4 and the [tex]\( y \)[/tex]-intercept [tex]\( b \)[/tex] is 3.
Let's analyze each given equation to find the one with the same slope (4) and correct [tex]\( y \)[/tex]-intercept.
### Option 1: [tex]\( y - 7 = 3(x - 1) \)[/tex]
This is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Slope [tex]\( m \)[/tex]: 3
- Since the slope is 3, it does not match the slope of 4.
### Option 2: [tex]\( y - 1 = 3(x - 7) \)[/tex]
This is also in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Slope [tex]\( m \)[/tex]: 3
- Again, the slope is 3, which does not match the slope of 4.
### Option 3: [tex]\( y - 7 = 4(x - 1) \)[/tex]
This is in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Slope [tex]\( m \)[/tex]: 4
- Here, the slope is 4, which matches the slope of our given line. Now we check the point:
- Point [tex]\((x_1, y_1)\)[/tex] is [tex]\((1, 7)\)[/tex], which lies on the line.
So, the equation [tex]\( y - 7 = 4(x - 1) \)[/tex] has the correct slope and passes through the given point [tex]\((1, 7)\)[/tex]. It represents the same line.
### Option 4: [tex]\( y - 1 = 4(x - 7) \)[/tex]
This is also in point-slope form [tex]\( y - y_1 = m(x - x_1) \)[/tex].
- Slope [tex]\( m \)[/tex]: 4
- The slope is 4, which matches, but we need to check the point:
- Point [tex]\((x_1, y_1)\)[/tex] is [tex]\((7, 1)\)[/tex]. When we convert this to slope-intercept form, it results in a different line.
Based on the analysis, the correct option that represents the same line as [tex]\( f(x) = 4x + 3 \)[/tex] is:
[tex]\[ y - 7 = 4(x - 1) \][/tex]
This corresponds to Option 3.
