Answer :
Let's analyze the function [tex]\( y = -x^2 + 3x + 8 \)[/tex].
1. Identify the type of function: This is a quadratic function because the highest power of [tex]\( x \)[/tex] is 2.
2. Identify the coefficients:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( -1 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex].
- The constant term is [tex]\( 8 \)[/tex].
3. Determine the vertex:
- The vertex form of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be used to find the vertex [tex]\((h, k)\)[/tex] where
[tex]\[ h = -\frac{b}{2a} \][/tex]
and
[tex]\[ k = y(h). \][/tex]
Here, [tex]\( a = -1 \)[/tex] and [tex]\( b = 3 \)[/tex], so:
[tex]\[ h = -\frac{3}{2 \times -1} = \frac{3}{2} = 1.5. \][/tex]
Plug [tex]\( h \)[/tex] back into the function to find [tex]\( k \)[/tex]:
[tex]\[ k = -\left(\frac{3}{2}\right)^2 + 3\left(\frac{3}{2}\right) + 8 = -\frac{9}{4} + \frac{9}{2} + 8. \][/tex]
Simplify this step by step:
[tex]\[ -\frac{9}{4} + \frac{9}{2} = -\frac{9}{4} + \frac{18}{4} = \frac{9}{4}, \][/tex]
[tex]\[ \frac{9}{4} + 8 = \frac{9}{4} + \frac{32}{4} = \frac{41}{4} = 10.25. \][/tex]
So, the vertex is [tex]\( \left( 1.5, 10.25 \right) \)[/tex].
4. Determine the axis of symmetry:
- The axis of symmetry of the parabola is the vertical line that passes through the vertex:
[tex]\[ x = 1.5. \][/tex]
5. Determine the direction of the parabola:
- Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( -1 \)[/tex]) is negative, the parabola opens downward.
6. Identify the y-intercept:
- The y-intercept is the point where the function crosses the [tex]\( y \)[/tex]-axis, i.e., when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = - (0)^2 + 3(0) + 8 = 8. \][/tex]
Thus, the y-intercept is [tex]\( (0, 8) \)[/tex].
7. Finding the x-intercepts:
- To find the x-intercepts, solve the equation [tex]\( y = 0 \)[/tex]:
[tex]\[ -x^2 + 3x + 8 = 0. \][/tex]
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 8 \)[/tex]:
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(8)}}{-2}. \][/tex]
Simplify:
[tex]\[ x = \frac{-3 \pm \sqrt{9 + 32}}{-2} = \frac{-3 \pm \sqrt{41}}{-2}. \][/tex]
Therefore, the x-intercepts are:
[tex]\[ x = \frac{-3 + \sqrt{41}}{-2} \quad \text{and} \quad x = \frac{-3 - \sqrt{41}}{-2}. \][/tex]
In summary:
- The function [tex]\( y = -x^2 + 3x + 8 \)[/tex] is a quadratic function.
- It has a vertex at [tex]\( (1.5, 10.25) \)[/tex].
- The axis of symmetry is [tex]\( x = 1.5 \)[/tex].
- The parabola opens downward.
- The y-intercept is [tex]\( (0, 8) \)[/tex].
- The x-intercepts are [tex]\(\frac{-3 + \sqrt{41}}{-2}\)[/tex] and [tex]\(\frac{-3 - \sqrt{41}}{-2}\)[/tex].
1. Identify the type of function: This is a quadratic function because the highest power of [tex]\( x \)[/tex] is 2.
2. Identify the coefficients:
- The coefficient of [tex]\( x^2 \)[/tex] is [tex]\( -1 \)[/tex].
- The coefficient of [tex]\( x \)[/tex] is [tex]\( 3 \)[/tex].
- The constant term is [tex]\( 8 \)[/tex].
3. Determine the vertex:
- The vertex form of a quadratic function [tex]\( y = ax^2 + bx + c \)[/tex] can be used to find the vertex [tex]\((h, k)\)[/tex] where
[tex]\[ h = -\frac{b}{2a} \][/tex]
and
[tex]\[ k = y(h). \][/tex]
Here, [tex]\( a = -1 \)[/tex] and [tex]\( b = 3 \)[/tex], so:
[tex]\[ h = -\frac{3}{2 \times -1} = \frac{3}{2} = 1.5. \][/tex]
Plug [tex]\( h \)[/tex] back into the function to find [tex]\( k \)[/tex]:
[tex]\[ k = -\left(\frac{3}{2}\right)^2 + 3\left(\frac{3}{2}\right) + 8 = -\frac{9}{4} + \frac{9}{2} + 8. \][/tex]
Simplify this step by step:
[tex]\[ -\frac{9}{4} + \frac{9}{2} = -\frac{9}{4} + \frac{18}{4} = \frac{9}{4}, \][/tex]
[tex]\[ \frac{9}{4} + 8 = \frac{9}{4} + \frac{32}{4} = \frac{41}{4} = 10.25. \][/tex]
So, the vertex is [tex]\( \left( 1.5, 10.25 \right) \)[/tex].
4. Determine the axis of symmetry:
- The axis of symmetry of the parabola is the vertical line that passes through the vertex:
[tex]\[ x = 1.5. \][/tex]
5. Determine the direction of the parabola:
- Since the coefficient of [tex]\( x^2 \)[/tex] (which is [tex]\( -1 \)[/tex]) is negative, the parabola opens downward.
6. Identify the y-intercept:
- The y-intercept is the point where the function crosses the [tex]\( y \)[/tex]-axis, i.e., when [tex]\( x = 0 \)[/tex]:
[tex]\[ y = - (0)^2 + 3(0) + 8 = 8. \][/tex]
Thus, the y-intercept is [tex]\( (0, 8) \)[/tex].
7. Finding the x-intercepts:
- To find the x-intercepts, solve the equation [tex]\( y = 0 \)[/tex]:
[tex]\[ -x^2 + 3x + 8 = 0. \][/tex]
Use the quadratic formula [tex]\( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \)[/tex], where [tex]\( a = -1 \)[/tex], [tex]\( b = 3 \)[/tex], and [tex]\( c = 8 \)[/tex]:
[tex]\[ x = \frac{-3 \pm \sqrt{3^2 - 4(-1)(8)}}{-2}. \][/tex]
Simplify:
[tex]\[ x = \frac{-3 \pm \sqrt{9 + 32}}{-2} = \frac{-3 \pm \sqrt{41}}{-2}. \][/tex]
Therefore, the x-intercepts are:
[tex]\[ x = \frac{-3 + \sqrt{41}}{-2} \quad \text{and} \quad x = \frac{-3 - \sqrt{41}}{-2}. \][/tex]
In summary:
- The function [tex]\( y = -x^2 + 3x + 8 \)[/tex] is a quadratic function.
- It has a vertex at [tex]\( (1.5, 10.25) \)[/tex].
- The axis of symmetry is [tex]\( x = 1.5 \)[/tex].
- The parabola opens downward.
- The y-intercept is [tex]\( (0, 8) \)[/tex].
- The x-intercepts are [tex]\(\frac{-3 + \sqrt{41}}{-2}\)[/tex] and [tex]\(\frac{-3 - \sqrt{41}}{-2}\)[/tex].
