What is the slope and the y-intercept?

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Understand the Problem

The question asks for the slope and y-intercept of a linear function represented by a table of values. This requires calculating the slope using the differences in the y-values and x-values and finding the y-intercept based on the line equation.

Answer

Slope = 4, y-intercept = -8
Answer for screen readers
  • Slope = 4
  • y-intercept = -8

Steps to Solve

  1. Identify points from the table The given points from the table are:

    • When $x = -2$, $y = -16$
    • When $x = 2$, $y = 0$
    • When $x = 4$, $y = 8$
    • When $x = 6$, $y = 16$
  2. Calculate the slope The formula for slope $m$ between two points $(x_1, y_1)$ and $(x_2, y_2)$ is given by:

    $$ m = \frac{y_2 - y_1}{x_2 - x_1} $$

    Using the points $(2, 0)$ and $(4, 8)$:

    $$ m = \frac{8 - 0}{4 - 2} = \frac{8}{2} = 4 $$

  3. Verify slope with another pair of points Let's also verify using the points $(2, 0)$ and $(6, 16)$:

    $$ m = \frac{16 - 0}{6 - 2} = \frac{16}{4} = 4 $$

    The slope is consistent and equals to 4.

  4. Determine the y-intercept The y-intercept occurs when $x = 0$. We can use the equation of the line in slope-intercept form, $y = mx + b$.

    Using the slope $m = 4$ and the point $(2, 0)$:

    Substitute $x = 2$ and $y = 0$:

    $$ 0 = 4(2) + b \implies 0 = 8 + b \implies b = -8 $$

    Therefore, the y-intercept is $-8$.

  • Slope = 4
  • y-intercept = -8

More Information

The slope indicates the rate of change of $y$ with respect to $x$. A slope of 4 means for every increase of 1 unit in $x$, $y$ increases by 4 units. The y-intercept, $-8$, tells us where the line crosses the $y$-axis.

Tips

  • Not using distinct points: Ensure you are using two different points to calculate the slope.
  • Incorrect calculations: Double-check subtraction and division steps when calculating slope and intercept.
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