Certainly! Let's tackle each part methodically, starting with parts (a)-(d) for the functions [tex]\( f(x) = -x^2 + 5 \)[/tex] and [tex]\( g(x) = -2x + 2 \)[/tex].
### Part (a)
Graph the functions [tex]\( f(x) \)[/tex] and [tex]\( g(x) \)[/tex] on the same Cartesian plane.
To determine the correct graph, let's understand how each function behaves:
1. Function [tex]\( f(x) = -x^2 + 5 \)[/tex]:
- This is a quadratic function, specifically a downward-facing parabola because the coefficient of [tex]\( x^2 \)[/tex] is negative.
- The vertex of this parabola is at [tex]\( (0, 5) \)[/tex] because it's in the form [tex]\( ax^2 + bx + c \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 0 \)[/tex], and [tex]\( c = 5 \)[/tex].
- To find more points, evaluate [tex]\( f(x) \)[/tex] at several values of [tex]\( x \)[/tex]:
- [tex]\( f(0) = -0^2 + 5 = 5 \)[/tex]
- [tex]\( f(1) = -1^2 + 5 = 4 \)[/tex]
- [tex]\( f(-1) = -(-1)^2 + 5 = 4 \)[/tex]
- It continues to decrease as [tex]\( |x| \)[/tex] increases.
2. Function [tex]\( g(x) = -2x + 2 \)[/tex]:
- This is a linear function with a slope of -2 and a y-intercept at 2.
- The y-intercept means the graph will cross the y-axis at (0, 2).
- The slope of -2 indicates that for every one unit increase in [tex]\( x \)[/tex], [tex]\( g(x) \)[/tex] decreases by 2 units.
- To find more points, evaluate [tex]\( g(x) \)[/tex] at several values of [tex]\( x \)[/tex]:
- [tex]\( g(0) = -2(0) + 2 = 2 \)[/tex]
- [tex]\( g(1) = -2(1) + 2 = 0 \)[/tex]
- [tex]\( g(-1) = -2(-1) + 2 = 4 \)[/tex]
### Sample Values Confirmed:
- For [tex]\( x = -10 \)[/tex]:
- [tex]\( f(-10) = -(-10)^2 + 5 = -100 + 5 = -95 \)[/tex]
- [tex]\( g(-10) = -2(-10) + 2 = 20 + 2 = 22 \)[/tex]
Using the same values:
- For [tex]\( x = 0 \)[/tex]:
- [tex]\( f(0) = -0^2 + 5 = 5 \)[/tex]
- [tex]\( g(0) = -2(0) + 2 = 2 \)[/tex]
- For [tex]\( x = 1 \)[/tex]:
- [tex]\( f(1) = -1^2 + 5 = 4 \)[/tex]
- [tex]\( g(1) = -2(1) + 2 = 0 \)[/tex]
### Choosing the Correct Graph:
Since I cannot see the graph options labeled as A, B, C, and D, the correct graph should have the following characteristics based on our analysis:
- A parabola with its vertex at (0, 5) opening downwards.
- A straight line crossing the y-axis at (0, 2) with a slope of -2.
Using these properties, you should be able to identify the correct graph labeled as the correct option from the provided choices.
If you have specific coordinates or more points to match within the given graphs, ensure they align with the behavior of both functions as we've described.
