Consider the linear function that is represented by the equation y = negative 10 x + 6 and the linear function that is represented by the equation y minus 36 = 8 (x minus 4). Which statement is correct regarding their slopes and y-intercepts?

The function that is represented by the equation y = negative 10 x + 6 has a steeper slope and a greater y-intercept.
The function that is represented by the equation y = negative 10 x + 6 has a steeper slope, and the function that is represented by the equation y minus 36 = 8 (x minus 4) has a greater y-intercept.
The function that is represented by the equation y minus 36 = 8 (x minus 4) has a steeper slope, and the function that is represented by the equation y = negative 10 x + 6 has a greater y-intercept.
The function that is represented by the equation y minus 36 = 8 (x minus 4) has a steeper slope and a greater y-intercept.

1 answer

To compare the slopes and y-intercepts of the two functions, we first rewrite both equations in the slope-intercept form (y = mx + b), where m represents the slope and b represents the y-intercept.

  1. The first equation is already in slope-intercept form: \( y = -10x + 6 \)

    • Slope (m) = -10
    • Y-intercept (b) = 6
  2. Now, let's rewrite the second equation \( y - 36 = 8(x - 4) \):

    • Distributing on the right side: \[ y - 36 = 8x - 32 \]
    • Adding 36 to both sides to solve for y: \[ y = 8x - 32 + 36 \] \[ y = 8x + 4 \]
    • Now we have:
      • Slope (m) = 8
      • Y-intercept (b) = 4

Now, we can compare the slopes and y-intercepts:

  • Slopes:

    • The slope of the first function is -10.
    • The slope of the second function is 8.
    • Since the absolute value of -10 (which is 10) is greater than 8, the first function has a steeper slope.
  • Y-intercepts:

    • The y-intercept of the first function is 6.
    • The y-intercept of the second function is 4.
    • Since 6 is greater than 4, the first function has a greater y-intercept.

Based on this comparison, the correct statement is:

The function that is represented by the equation \( y = -10x + 6 \) has a steeper slope and a greater y-intercept.