Which of the following quadratics will have a minimum value of -5 at $$ x=7 $$ ? $$ \begin{array}{ll} ext { (1) } y=x^{2}-14 x+39 & ext { (3) } y=x^{2}-14 x+44 \ ext { (2) } y=-x^{2}+14 x-54 & ext { (4) } y=-x^{2}-10 x-18\end{array} $$

Question

Which of the following quadratics will have a minimum value of -5 at $$ x=7 $$ ? $$ \begin{array}{ll}	ext { (1) } y=x^{2}-14 x+39 & 	ext { (3) } y=x^{2}-14 x+44 \ 	ext { (2) } y=-x^{2}+14 x-54 & 	ext { (4) } y=-x^{2}-10 x-18\end{array} $$

UpStudy AI · Accepted Answer

To determine which of the given quadratics has a minimum value of -5 at $$ x=7 $$, we need to analyze each quadratic function.

1. **Identify the general form of a quadratic function**: 
   A quadratic function can be expressed as $$ y = ax^2 + bx + c $$. The vertex form of a quadratic is given by $$ y = a(x-h)^2 + k $$, where $$ (h, k) $$ is the vertex of the parabola.

2. **Find the vertex for each quadratic**: 
   The vertex $$ x $$ coordinate can be found using the formula $$ h = -\frac{b}{2a} $$. We will also evaluate the function at $$ x=7 $$ to check if it equals -5.

Let's analyze each quadratic one by one.

### Quadratic (1): $$ y = x^2 - 14x + 39 $$

- Coefficients: $$ a = 1, b = -14, c = 39 $$
- Vertex $$ x $$ coordinate: 
  $$
  h = -\frac{-14}{2 \cdot 1} = 7
  $$
- Evaluate at $$ x = 7 $$:
  $$
  y(7) = 7^2 - 14 \cdot 7 + 39 = 49 - 98 + 39 = -10
  $$

### Quadratic (2): $$ y = -x^2 + 14x - 54 $$

- Coefficients: $$ a = -1, b = 14, c = -54 $$
- Vertex $$ x $$ coordinate: 
  $$
  h = -\frac{14}{2 \cdot -1} = 7
  $$
- Evaluate at $$ x = 7 $$:
  $$
  y(7) = -7^2 + 14 \cdot 7 - 54 = -49 + 98 - 54 = -5
  $$

### Quadratic (3): $$ y = x^2 - 14x + 44 $$

- Coefficients: $$ a = 1, b = -14, c = 44 $$
- Vertex $$ x $$ coordinate: 
  $$
  h = -\frac{-14}{2 \cdot 1} = 7
  $$
- Evaluate at $$ x = 7 $$:
  $$
  y(7) = 7^2 - 14 \cdot 7 + 44 = 49 - 98 + 44 = -5
  $$

### Quadratic (4): $$ y = -x^2 - 10x - 18 $$

- Coefficients: $$ a = -1, b = -10, c = -18 $$
- Vertex $$ x $$ coordinate: 
  $$
  h = -\frac{-10}{2 \cdot -1} = 5
  $$
- Evaluate at $$ x = 7 $$:
  $$
  y(7) = -7^2 - 10 \cdot 7 - 18 = -49 - 70 - 18 = -137
  $$

### Conclusion:
The quadratics that have a minimum value of -5 at $$ x=7 $$ are:

- Quadratic (2): $$ y = -x^2 + 14x - 54 $$
- Quadratic (3): $$ y = x^2 - 14x + 44 $$

Thus, the answer is quadratics (2) and (3).
Quadratics (2) and (3) have a minimum value of -5 at $$ x=7 $$.

Which of the following quadratics will have a minimum value of -5 at ?

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Solución

Quadratic (1):

Quadratic (2):

Quadratic (3):

Quadratic (4):

Conclusion:

Beyond the Answer

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