To graph the equation [tex]\( y = -x^2 + 10x - 16 \)[/tex] and plot 5 points including the roots and the vertex, we need to follow a step-by-step approach:
### Step 1: Identify the Roots
The roots of the quadratic equation can be found using the quadratic formula:
[tex]\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \][/tex]
For the equation [tex]\( y = -x^2 + 10x - 16 \)[/tex]:
- [tex]\( a = -1 \)[/tex]
- [tex]\( b = 10 \)[/tex]
- [tex]\( c = -16 \)[/tex]
Perform the calculations:
[tex]\[ b^2 - 4ac = 10^2 - 4(-1)(-16) = 100 - 64 = 36 \][/tex]
Thus,
[tex]\[ x = \frac{-10 \pm \sqrt{36}}{2(-1)} = \frac{-10 \pm 6}{-2} \][/tex]
Calculate the two roots:
[tex]\[ x_1 = \frac{-10 + 6}{-2} = \frac{-4}{-2} = 2 \][/tex]
[tex]\[ x_2 = \frac{-10 - 6}{-2} = \frac{-16}{-2} = 8 \][/tex]
So, the roots are [tex]\( (2, 0) \)[/tex] and [tex]\( (8, 0) \)[/tex].
### Step 2: Identify the Vertex
The vertex of a parabola [tex]\( y = ax^2 + bx + c \)[/tex] is found using the formula:
[tex]\[ x = -\frac{b}{2a} \][/tex]
For [tex]\( a = -1 \)[/tex] and [tex]\( b = 10 \)[/tex]:
[tex]\[ x = -\frac{10}{2(-1)} = 5 \][/tex]
To find the y-coordinate of the vertex:
[tex]\[ y = -5^2 + 10(5) - 16 = -25 + 50 - 16 = 9 \][/tex]
So, the vertex is [tex]\( (5, 9) \)[/tex].
### Step 3: Identify Additional Points
To plot additional points, choose [tex]\( x \)[/tex] values around the vertex.
For [tex]\( x = 4 \)[/tex]:
[tex]\[ y = -(4)^2 + 10(4) - 16 = -16 + 40 - 16 = 8 \][/tex]
Thus, the point is [tex]\( (4, 8) \)[/tex].
For [tex]\( x = 6 \)[/tex]:
[tex]\[ y = -(6)^2 + 10(6) - 16 = -36 + 60 - 16 = 8 \][/tex]
Thus, the point is [tex]\( (6, 8) \)[/tex].
### Summary of Points
1. Root 1: [tex]\( (2, 0) \)[/tex]
2. Root 2: [tex]\( (8, 0) \)[/tex]
3. Vertex: [tex]\( (5, 9) \)[/tex]
4. Additional Point 1: [tex]\( (4, 8) \)[/tex]
5. Additional Point 2: [tex]\( (6, 8) \)[/tex]
### Step 4: Plot these Points
Now that we've calculated these points, you can plot them on a set of axes. The parabola will open downwards because the coefficient of [tex]\( x^2 \)[/tex] is negative.
Here's a sketch of how the graph should look with these points:
1. Plot [tex]\( (2, 0) \)[/tex]
2. Plot [tex]\( (8, 0) \)[/tex]
3. Plot [tex]\( (5, 9) \)[/tex]
4. Plot [tex]\( (4, 8) \)[/tex]
5. Plot [tex]\( (6, 8) \)[/tex]
After plotting these points, draw a smooth curve through them to complete the graph of [tex]\( y = -x^2 + 10x - 16 \)[/tex].
