Find the Maximum/Minimum Value y=4(x-1)^2+3

$y = 4 {(x - 1)}^{2} + 3$

Step 1

Simplify each term.

Rewrite ${(x - 1)}^{2}$ as $(x - 1) (x - 1)$ .

$f (x) = 4 ((x - 1) (x - 1)) + 3$

Expand $(x - 1) (x - 1)$ using the FOIL Method.

Apply the distributive property.

$f (x) = 4 (x (x - 1) - 1 (x - 1)) + 3$

Apply the distributive property.

$f (x) = 4 (x \cdot x + x \cdot - 1 - 1 (x - 1)) + 3$

Apply the distributive property.

$f (x) = 4 (x \cdot x + x \cdot - 1 - 1 x - 1 \cdot - 1) + 3$

Simplify and combine like terms.

Simplify each term.

Multiply $x$ by $x$ .

$f (x) = 4 (x^{2} + x \cdot - 1 - 1 x - 1 \cdot - 1) + 3$

Move $- 1$ to the left of $x$ .

$f (x) = 4 (x^{2} - 1 \cdot x - 1 x - 1 \cdot - 1) + 3$

Rewrite $- 1 x$ as $- x$ .

$f (x) = 4 (x^{2} - x - 1 x - 1 \cdot - 1) + 3$

Rewrite $- 1 x$ as $- x$ .

$f (x) = 4 (x^{2} - x - x - 1 \cdot - 1) + 3$

Multiply $- 1$ by $- 1$ .

$f (x) = 4 (x^{2} - x - x + 1) + 3$

Subtract $x$ from $- x$ .

$f (x) = 4 (x^{2} - 2 x + 1) + 3$

Apply the distributive property.

$f (x) = 4 x^{2} + 4 (- 2 x) + 4 \cdot 1 + 3$

Simplify.

Multiply $- 2$ by $4$ .

$f (x) = 4 x^{2} - 8 x + 4 \cdot 1 + 3$

Multiply $4$ by $1$ .

$f (x) = 4 x^{2} - 8 x + 4 + 3$

Step 2

Add $4$ and $3$ .

$f (x) = 4 x^{2} - 8 x + 7$

Step 3

The minimum of a quadratic function occurs at $x = - \frac{b}{2 a}$ . If $a$ is positive, the minimum value of the function is $f (- \frac{b}{2 a})$ .

$f_{min}$ $x = a x^{2} + b x + c$ occurs at $x = - \frac{b}{2 a}$

Step 4

Find the value of $x = - \frac{b}{2 a}$ .

Substitute in the values of $a$ and $b$ .

$x = - \frac{- 8}{2 (4)}$

Remove parentheses.

$x = - \frac{- 8}{2 (4)}$

Simplify $- \frac{- 8}{2 (4)}$ .

Cancel the common factor of $- 8$ and $2$ .

Factor $2$ out of $- 8$ .

$x = - \frac{2 \cdot - 4}{2 \cdot 4}$

Cancel the common factors.

Factor $2$ out of $2 \cdot 4$ .

$x = - \frac{2 \cdot - 4}{2 (4)}$

Cancel the common factor.

$x = - \frac{2 \cdot - 4}{2 \cdot 4}$

Rewrite the expression.

$x = - \frac{- 4}{4}$

Cancel the common factor of $- 4$ and $4$ .

Factor $4$ out of $- 4$ .

$x = - \frac{4 \cdot - 1}{4}$

Cancel the common factors.

Factor $4$ out of $4$ .

$x = - \frac{4 \cdot - 1}{4 (1)}$

Cancel the common factor.

$x = - \frac{4 \cdot - 1}{4 \cdot 1}$

Rewrite the expression.

$x = - \frac{- 1}{1}$

Divide $- 1$ by $1$ .

$x = - - 1$

Multiply $- 1$ by $- 1$ .

$x = 1$

Step 5

Evaluate $f (1)$ .

Replace the variable $x$ with $1$ in the expression.

$f (1) = 4 {(1)}^{2} - 8 \cdot 1 + 7$

Simplify the result.

Simplify each term.

One to any power is one.

$f (1) = 4 \cdot 1 - 8 \cdot 1 + 7$

Multiply $4$ by $1$ .

$f (1) = 4 - 8 \cdot 1 + 7$

Multiply $- 8$ by $1$ .

$f (1) = 4 - 8 + 7$

Simplify by adding and subtracting.

Subtract $8$ from $4$ .

$f (1) = - 4 + 7$

Add $- 4$ and $7$ .

$f (1) = 3$

The final answer is $3$ .

$3$

Step 6

Use the $x$ and $y$ values to find where the minimum occurs.

$(1, 3)$

Step 7

Find the Maximum/Minimum Value y=4(x-1)^2+3

Related Questions

Support LCMGCF.com by disabling your adblocker.