Let's analyze the given functions step by step. The functions provided are:
[tex]\[ R(x) = -x^2 + 6x \][/tex]
[tex]\[ C(x) = x + 4 \][/tex]
To complete parts (a) through (d) of the problem, we need to identify specific features of these functions.
### Part (a): Graphing the Functions
To graph both functions, we need to understand their shapes and key features.
1. Graph of [tex]\( R(x) = -x^2 + 6x \)[/tex]:
- This is a downward-opening parabola (since the coefficient of [tex]\( x^2 \)[/tex] is negative).
2. Graph of [tex]\( C(x) = x + 4 \)[/tex]:
- This is a straight line with a slope of 1 and a y-intercept of 4.
### Part (b): Identify the vertex of [tex]\( R(x) \)[/tex]
The vertex of the parabola [tex]\( R(x) = -x^2 + 6x \)[/tex] can be found using the vertex formula for a quadratic function [tex]\( ax^2 + bx + c \)[/tex]:
The x-coordinate of the vertex is given by:
[tex]\[ x_{\text{vertex}} = -\frac{b}{2a} \][/tex]
For [tex]\( R(x) = -x^2 + 6x \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 6 \)[/tex]
So, we calculate:
[tex]\[ x_{\text{vertex}} = -\frac{6}{2 \cdot (-1)} = 3 \][/tex]
Next, we find the y-coordinate of the vertex by substituting [tex]\( x = 3 \)[/tex] back into the function [tex]\( R(x) \)[/tex]:
[tex]\[ R(3) = -(3)^2 + 6 \cdot 3 = -9 + 18 = 9 \][/tex]
Therefore, the vertex of [tex]\( R(x) \)[/tex] is at:
[tex]\[ (3, 9) \][/tex]
### Part (c): Identify the y-intercept of [tex]\( C(x) \)[/tex]
The y-intercept of [tex]\( C(x) \)[/tex] is found by evaluating the function at [tex]\( x = 0 \)[/tex]:
[tex]\[ C(0) = 0 + 4 = 4 \][/tex]
Thus, the y-intercept of [tex]\( C(x) \)[/tex] is at:
[tex]\[ y = 4 \][/tex]
### Summary of Results:
- The vertex of [tex]\( R(x) \)[/tex] is at [tex]\( (3, 9) \)[/tex].
- The y-intercept of [tex]\( C(x) \)[/tex] is at [tex]\( y = 4 \)[/tex].
