1. The function $$ f $$ defined by $$ y=f(x)=-x^{2}+6 x-5 $$ has (A) A minimum $$ y $$ value and a negative $$ y $$-intercept. (B) A maximum $$ y $$ value and a positive $$ y $$-intercept. (C) A minimum $$ y $$ value and a positive $$ y $$-intercept. (D) A maximum $$ y $$ value and a negative $$ y $$-intercept. Under the heading Algebraic Processes one of the topics listed is Algebraic Manipulation. Quest is an example of a question where the answer cannot be deduced by substituting into the given options to rule out those that are correct.

Question

UpStudy AI · Accepted Answer

To determine the correct answer for the function $$ f(x) = -x^{2} + 6x - 5 $$, let's analyze its properties step by step.

### 1. **Nature of the Parabola**
The given function is a quadratic function of the form:
$$
f(x) = ax^{2} + bx + c
$$
where:
- $$ a = -1 $$ (negative)
- $$ b = 6 $$
- $$ c = -5 $$

Since the coefficient $$ a $$ is negative ($$ a = -1 $$), the parabola opens **downwards**. This implies that the function has a **maximum** value, not a minimum.

### 2. **Y-Intercept**
The y-intercept of a function occurs where $$ x = 0 $$. Substituting $$ x = 0 $$ into the function:
$$
f(0) = -0^{2} + 6(0) - 5 = -5
$$
So, the y-intercept is **-5**, which is **negative**.

### 3. **Conclusion**
Based on the analysis:
- The function has a **maximum y-value**.
- The y-intercept is **negative**.

### **Correct Option:**
**(D) A maximum $$ y $$ value and a negative $$ y $$-intercept.**

### **Why Algebraic Manipulation is Necessary**
In this problem, simple substitution into the answer choices isn't sufficient to identify the correct option. Instead, understanding the properties of quadratic functions—such as the direction of the parabola (determined by the coefficient $$ a $$) and calculating specific values like the y-intercept—is essential. This requires **algebraic manipulation** and analytical reasoning rather than mere substitution, aligning with the topic of Algebraic Manipulation under Algebraic Processes.
The function $$ f(x) = -x^{2} + 6x - 5 $$ has a **maximum y-value** and a **negative y-intercept**. Therefore, the correct answer is: **(D) A maximum $$ y $$ value and a negative $$ y $$-intercept.**

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Solución

1. Nature of the Parabola

2. Y-Intercept

3. Conclusion

Correct Option:

Why Algebraic Manipulation is Necessary

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