Determine whether the quadratic function shown below has a minimum or maximum, then
determine the minimum or maximum value of the function.

Question

Determine whether the quadratic function shown below has a minimum or maximum, then determine the minimum or maximum value of the function. $$ f(x)=(x+5)^{2}-2 $$

UpStudy AI · Accepted Answer

The given quadratic function is:

$$ f(x) = (x + 5)^2 - 2 $$

### Determining Minimum or Maximum

Quadratic functions can be written in the form:

$$ f(x) = a(x - h)^2 + k $$

where:
- $$ a $$ determines the direction of the parabola.
  - If $$ a > 0 $$, the parabola opens **upwards**, and the function has a **minimum** value.
  - If $$ a < 0 $$, the parabola opens **downwards**, and the function has a **maximum** value.

In the given function:

$$ f(x) = (x + 5)^2 - 2 $$

- The coefficient $$ a = 1 $$ (since it's the coefficient of $$(x + 5)^2$$).
- Since $$ a = 1 > 0 $$, the parabola opens **upwards**.

**Conclusion:** The function has a **minimum** value.

### Determining the Minimum Value

The function is already in vertex form:

$$ f(x) = (x + 5)^2 - 2 $$

In the vertex form $$ f(x) = a(x - h)^2 + k $$, the vertex is at the point $$ (h, k) $$.

For the given function:
- $$ h = -5 $$
- $$ k = -2 $$

**Conclusion:** The minimum value of the function is **-2**, occurring at $$ x = -5 $$.

### Final Answer

**The quadratic function has a minimum value of –2.**
The quadratic function has a minimum value of –2.

Determine whether the quadratic function shown below has a minimum or maximum, then
determine the minimum or maximum value of the function.

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