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Build the equation that combines the functions b=3r+1

and y=2b2
.(2 points)
y=
r2+
r+

1 answer

To combine the functions \( b = 3r + 1 \) and \( y = 2b^2 \), we need to substitute the expression for \( b \) into the equation for \( y \).

  1. Start by substituting \( b \) in the equation for \( y \): \[ y = 2b^2 \] Substituting \( b = 3r + 1 \): \[ y = 2(3r + 1)^2 \]

  2. Now, we need to expand \( (3r + 1)^2 \): \[ (3r + 1)^2 = (3r)^2 + 2(3r)(1) + (1)^2 = 9r^2 + 6r + 1 \]

  3. Substitute this back into the \( y \) equation: \[ y = 2(9r^2 + 6r + 1) \]

  4. Distribute the 2: \[ y = 18r^2 + 12r + 2 \]

Now we have the combined equation: \[ y = 18r^2 + 12r + 2 \]

So, the final equation is: \[ y = 18r^2 + 12r + 2 \]