To graph the equation [tex]\( y = -x^2 + 10x - 24 \)[/tex], we need to identify key points on the graph, specifically the roots (where the function crosses the x-axis) and the vertex (the highest or lowest point of the parabola). Additionally, we will choose two more points to ensure an accurate graph. Here are the steps:
### 1. Find the Roots (x-intercepts)
The roots of the equation [tex]\( y = -x^2 + 10x - 24 \)[/tex] are the values of [tex]\( x \)[/tex] where [tex]\( y = 0 \)[/tex]. Solving the equation:
1. [tex]\( x_1 = 6.000000000000001 \)[/tex]
2. [tex]\( x_2 = 3.9999999999999996 \)[/tex]
So, the roots are approximately at [tex]\( x = 6 \)[/tex] and [tex]\( x = 4 \)[/tex].
### 2. Find the Vertex
The vertex of a parabola in the form [tex]\( y = ax^2 + bx + c \)[/tex] can be found using the formula [tex]\( x = -\frac{b}{2a} \)[/tex]. For our equation [tex]\( y = -x^2 + 10x - 24 \)[/tex]:
[tex]\[ a = -1, \quad b = 10 \][/tex]
[tex]\[ x = -\frac{10}{2(-1)} = 5 \][/tex]
To find the [tex]\( y \)[/tex]-coordinate of the vertex, substitute [tex]\( x = 5 \)[/tex] back into the equation:
[tex]\[ y = -(5)^2 + 10(5) - 24 \][/tex]
[tex]\[ y = -25 + 50 - 24 \][/tex]
[tex]\[ y = 1 \][/tex]
So, the vertex is at [tex]\( (5, 1) \)[/tex].
### 3. Select Additional Points
To get a complete picture of the parabola, we need two additional points. We choose [tex]\( x = 5.0 \)[/tex] and [tex]\( x = 5.000000000000001 \)[/tex]:
1. When [tex]\( x = 5.0 \)[/tex], [tex]\( y = 1.0 \)[/tex].
2. When [tex]\( x = 5.000000000000001 \)[/tex], [tex]\( y = 1.0 \)[/tex].
### 4. Plot the Points
Now, we'll plot the points:
1. [tex]\( (6.000000000000001, -7.105427357601002e-15) \)[/tex]
2. [tex]\( (3.9999999999999996, -3.552713678800501e-15) \)[/tex]
3. [tex]\( (5, 1) \)[/tex]
4. [tex]\( (5.000000000000001, 1.0) \)[/tex]
5. [tex]\( (5.0, 1.0) \)[/tex]
### Final Step: Draw the Graph
- The points are slightly approximate due to floating-point representation but you can treat them as:
- [tex]\( (6, 0) \)[/tex]
- [tex]\( (4, 0) \)[/tex]
- [tex]\( (5, 1) \)[/tex]
- [tex]\( (5, 1) \)[/tex]
- [tex]\( (5, 1) \)[/tex]
- These points should be marked on your graph. Draw a smooth parabolic curve through these points to represent the equation [tex]\( y = -x^2 + 10x - 24 \)[/tex].
This will give you a clear representation of the parabola with the identified roots and vertex.
