Answer :
To graph the equation [tex]\( y = -x^2 + 10x - 24 \)[/tex], we need to follow several steps: finding the vertex, determining the roots, and identifying additional points for plotting. Here's the step-by-step process:
1. Find the Vertex:
The vertex form of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -1 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = -24 \)[/tex].
- Calculating the x-coordinate of the vertex:
[tex]\[ x = -\frac{10}{2(-1)} = 5 \][/tex]
- Substitute [tex]\( x = 5 \)[/tex] back into the equation to find the y-coordinate:
[tex]\[ y = -(5)^2 + 10(5) - 24 = -25 + 50 - 24 = 1 \][/tex]
- So, the vertex is at [tex]\( (5, 1) \)[/tex].
2. Find the Roots (x-intercepts):
The roots of the quadratic equation are found by setting [tex]\( y = 0 \)[/tex] and solving for [tex]\( x \)[/tex]:
[tex]\[ -x^2 + 10x - 24 = 0 \][/tex]
- This can be solved using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{-10 \pm \sqrt{10^2 - 4(-1)(-24)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{-10 \pm \sqrt{100 - 96}}{-2} \][/tex]
[tex]\[ x = \frac{-10 \pm 2}{-2} \][/tex]
- Solving for the two roots:
[tex]\[ x_1 = \frac{-10 + 2}{-2} = 4 \][/tex]
[tex]\[ x_2 = \frac{-10 - 2}{-2} = 6 \][/tex]
- So, the roots are at [tex]\( x_1 = 4 \)[/tex] and [tex]\( x_2 = 6 \)[/tex].
3. Determine Additional Points:
We need at least two additional points to plot the curve properly. We can select [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex]. Substituting these values into the quadratic equation:
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2)^2 + 10(2) - 24 = -4 + 20 - 24 = -8 \][/tex]
- When [tex]\( x = 8 \)[/tex]:
[tex]\[ y = -(8)^2 + 10(8) - 24 = -64 + 80 - 24 = -8 \][/tex]
- So, the points are [tex]\( (2, -8) \)[/tex] and [tex]\( (8, -8) \)[/tex].
4. Plot the Points and Graph the Equation:
To summarize, we'll plot the following points:
- Vertex: [tex]\( (5, 1) \)[/tex]
- Roots: [tex]\( (4, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex]
- Additional points: [tex]\( (2, -8) \)[/tex] and [tex]\( (8, -8) \)[/tex]
### Graph:
1. Plot each of these points on the coordinate grid.
2. Draw a smooth curve through these points to represent the parabola.
#### Coordinates:
- Vertex: [tex]\( (5, 1) \)[/tex]
- Roots: [tex]\( (4, 0) \)[/tex], [tex]\( (6, 0) \)[/tex]
- Additional points: [tex]\( (2, -8) \)[/tex], [tex]\( (8, -8) \)[/tex]
By plotting these points accurately and drawing a smooth curve through them, you will have the graph of the quadratic equation [tex]\( y = -x^2 + 10x - 24 \)[/tex].
1. Find the Vertex:
The vertex form of a quadratic equation [tex]\( y = ax^2 + bx + c \)[/tex] is given by the formula [tex]\( x = -\frac{b}{2a} \)[/tex].
- Here, [tex]\( a = -1 \)[/tex], [tex]\( b = 10 \)[/tex], and [tex]\( c = -24 \)[/tex].
- Calculating the x-coordinate of the vertex:
[tex]\[ x = -\frac{10}{2(-1)} = 5 \][/tex]
- Substitute [tex]\( x = 5 \)[/tex] back into the equation to find the y-coordinate:
[tex]\[ y = -(5)^2 + 10(5) - 24 = -25 + 50 - 24 = 1 \][/tex]
- So, the vertex is at [tex]\( (5, 1) \)[/tex].
2. Find the Roots (x-intercepts):
The roots of the quadratic equation are found by setting [tex]\( y = 0 \)[/tex] and solving for [tex]\( x \)[/tex]:
[tex]\[ -x^2 + 10x - 24 = 0 \][/tex]
- This can be solved using the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex]:
[tex]\[ x = \frac{-10 \pm \sqrt{10^2 - 4(-1)(-24)}}{2(-1)} \][/tex]
[tex]\[ x = \frac{-10 \pm \sqrt{100 - 96}}{-2} \][/tex]
[tex]\[ x = \frac{-10 \pm 2}{-2} \][/tex]
- Solving for the two roots:
[tex]\[ x_1 = \frac{-10 + 2}{-2} = 4 \][/tex]
[tex]\[ x_2 = \frac{-10 - 2}{-2} = 6 \][/tex]
- So, the roots are at [tex]\( x_1 = 4 \)[/tex] and [tex]\( x_2 = 6 \)[/tex].
3. Determine Additional Points:
We need at least two additional points to plot the curve properly. We can select [tex]\( x = 2 \)[/tex] and [tex]\( x = 8 \)[/tex]. Substituting these values into the quadratic equation:
- When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2)^2 + 10(2) - 24 = -4 + 20 - 24 = -8 \][/tex]
- When [tex]\( x = 8 \)[/tex]:
[tex]\[ y = -(8)^2 + 10(8) - 24 = -64 + 80 - 24 = -8 \][/tex]
- So, the points are [tex]\( (2, -8) \)[/tex] and [tex]\( (8, -8) \)[/tex].
4. Plot the Points and Graph the Equation:
To summarize, we'll plot the following points:
- Vertex: [tex]\( (5, 1) \)[/tex]
- Roots: [tex]\( (4, 0) \)[/tex] and [tex]\( (6, 0) \)[/tex]
- Additional points: [tex]\( (2, -8) \)[/tex] and [tex]\( (8, -8) \)[/tex]
### Graph:
1. Plot each of these points on the coordinate grid.
2. Draw a smooth curve through these points to represent the parabola.
#### Coordinates:
- Vertex: [tex]\( (5, 1) \)[/tex]
- Roots: [tex]\( (4, 0) \)[/tex], [tex]\( (6, 0) \)[/tex]
- Additional points: [tex]\( (2, -8) \)[/tex], [tex]\( (8, -8) \)[/tex]
By plotting these points accurately and drawing a smooth curve through them, you will have the graph of the quadratic equation [tex]\( y = -x^2 + 10x - 24 \)[/tex].
