(2) The graph of a linear function passes through the points \( (3,19) \) and \( (5,23) \). Write an equation for the function. Show your work.
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To find the equation of the linear function that passes through the points \( (3,19) \) and \( (5,23) \), we can start by calculating the slope \( m \). The formula for slope between two points \( (x_1, y_1) \) and \( (x_2, y_2) \) is: \[ m = \frac{y_2 - y_1}{x_2 - x_1} \] Substituting in our points, we have: \[ m = \frac{23 - 19}{5 - 3} = \frac{4}{2} = 2 \] Now that we have the slope, we can use the point-slope form of the equation, which is: \[ y - y_1 = m(x - x_1) \] Using point \( (3, 19) \): \[ y - 19 = 2(x - 3) \] Now, simplifying this: \[ y - 19 = 2x - 6 \\ y = 2x + 13 \] Thus, the equation of the function is: \[ y = 2x + 13 \] Now you can plot this line, and it will beautifully intersect your given points!