Answer :
Let's systematically solve the provided quadratic function [tex]\( f(x) = 3x^2 - 18x + 24 \)[/tex] step-by-step:
### a) Finding the y-intercept:
The y-intercept of a function is the point at which the graph crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3(0)^2 - 18(0) + 24 = 24 \][/tex]
So, the y-intercept is:
[tex]\[ y = 24 \][/tex]
### b) Finding the x-intercepts:
The x-intercepts of a function are the points at which the graph crosses the x-axis. This occurs when [tex]\( f(x) = 0 \)[/tex], i.e., we need to solve the quadratic equation:
[tex]\[ 3x^2 - 18x + 24 = 0 \][/tex]
We can use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Where [tex]\( a = 3 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = 24 \)[/tex]. First, compute the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-18)^2 - 4(3)(24) \][/tex]
[tex]\[ \Delta = 324 - 288 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
Since the discriminant is positive, there are two real roots:
[tex]\[ x_1 = \frac{{-b + \sqrt{\Delta}}}{2a} \][/tex]
[tex]\[ x_1 = \frac{{18 + \sqrt{36}}}{6} \][/tex]
[tex]\[ x_1 = \frac{{18 + 6}}{6} \][/tex]
[tex]\[ x_1 = 4 \][/tex]
[tex]\[ x_2 = \frac{{-b - \sqrt{\Delta}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{{18 - \sqrt{36}}}{6} \][/tex]
[tex]\[ x_2 = \frac{{18 - 6}}{6} \][/tex]
[tex]\[ x_2 = 2 \][/tex]
So, the x-intercepts are:
[tex]\[ x = 4 \][/tex] and [tex]\[ x = 2 \][/tex]
### c) Finding the equation of the axis of symmetry:
The axis of symmetry for a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a = 3 \)[/tex] and [tex]\( b = -18 \)[/tex]:
[tex]\[ x = -\frac{-18}{2(3)} \][/tex]
[tex]\[ x = \frac{18}{6} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the equation of the axis of symmetry is:
[tex]\[ x = 3 \][/tex]
### d) Finding the vertex:
The vertex form of a quadratic function [tex]\( f(x) = a(x-h)^2 + k \)[/tex] and the vertex [tex]\((h, k)\)[/tex] can be found using the point on the axis of symmetry.
We already found [tex]\( x = 3 \)[/tex] (the x-coordinate of the vertex). Now, substitute [tex]\( x = 3 \)[/tex] into the original function to find the y-coordinate:
[tex]\[ f(3) = 3(3)^2 - 18(3) + 24 \][/tex]
[tex]\[ f(3) = 3(9) - 54 + 24 \][/tex]
[tex]\[ f(3) = 27 - 54 + 24 \][/tex]
[tex]\[ f(3) = -3 \][/tex]
So, the vertex is:
[tex]\[ (3, -3) \][/tex]
### e) Sketching the graph of the function:
To sketch the graph of [tex]\( f(x) = 3x^2 - 18x + 24 \)[/tex]:
1. Identify and plot the y-intercept: [tex]\((0, 24)\)[/tex]
2. Identify and plot the x-intercepts: [tex]\((2, 0)\)[/tex] and [tex]\((4, 0)\)[/tex]
3. Identify and plot the vertex: [tex]\((3, -3)\)[/tex]
4. Draw the axis of symmetry: Vertical line through [tex]\(x = 3\)[/tex]
The graph is a parabola opening upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive) with the vertex as the lowest point.
### Graph:
```
|
25| X
20|
15|
10|
5|
0|----X--------X-------------------
2 3 4
-5| X Vertex(3, -3)
///(Parabola sketch)
```
Label key points:
- [tex]\( (0, 24) \)[/tex] as y-intercept
- [tex]\( (2, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex] as x-intercepts
- [tex]\( (3, -3) \)[/tex] as vertex, with the axis of symmetry as [tex]\( x = 3 \)[/tex]
This careful plotting covers all necessary points and shapes required for an effective sketch.
### a) Finding the y-intercept:
The y-intercept of a function is the point at which the graph crosses the y-axis. This occurs when [tex]\( x = 0 \)[/tex].
[tex]\[ f(0) = 3(0)^2 - 18(0) + 24 = 24 \][/tex]
So, the y-intercept is:
[tex]\[ y = 24 \][/tex]
### b) Finding the x-intercepts:
The x-intercepts of a function are the points at which the graph crosses the x-axis. This occurs when [tex]\( f(x) = 0 \)[/tex], i.e., we need to solve the quadratic equation:
[tex]\[ 3x^2 - 18x + 24 = 0 \][/tex]
We can use the quadratic formula:
[tex]\[ x = \frac{{-b \pm \sqrt{{b^2 - 4ac}}}}{2a} \][/tex]
Where [tex]\( a = 3 \)[/tex], [tex]\( b = -18 \)[/tex], and [tex]\( c = 24 \)[/tex]. First, compute the discriminant ([tex]\( \Delta \)[/tex]):
[tex]\[ \Delta = b^2 - 4ac \][/tex]
[tex]\[ \Delta = (-18)^2 - 4(3)(24) \][/tex]
[tex]\[ \Delta = 324 - 288 \][/tex]
[tex]\[ \Delta = 36 \][/tex]
Since the discriminant is positive, there are two real roots:
[tex]\[ x_1 = \frac{{-b + \sqrt{\Delta}}}{2a} \][/tex]
[tex]\[ x_1 = \frac{{18 + \sqrt{36}}}{6} \][/tex]
[tex]\[ x_1 = \frac{{18 + 6}}{6} \][/tex]
[tex]\[ x_1 = 4 \][/tex]
[tex]\[ x_2 = \frac{{-b - \sqrt{\Delta}}}{2a} \][/tex]
[tex]\[ x_2 = \frac{{18 - \sqrt{36}}}{6} \][/tex]
[tex]\[ x_2 = \frac{{18 - 6}}{6} \][/tex]
[tex]\[ x_2 = 2 \][/tex]
So, the x-intercepts are:
[tex]\[ x = 4 \][/tex] and [tex]\[ x = 2 \][/tex]
### c) Finding the equation of the axis of symmetry:
The axis of symmetry for a quadratic function in the form [tex]\( ax^2 + bx + c \)[/tex] is given by:
[tex]\[ x = -\frac{b}{2a} \][/tex]
Substitute [tex]\( a = 3 \)[/tex] and [tex]\( b = -18 \)[/tex]:
[tex]\[ x = -\frac{-18}{2(3)} \][/tex]
[tex]\[ x = \frac{18}{6} \][/tex]
[tex]\[ x = 3 \][/tex]
So, the equation of the axis of symmetry is:
[tex]\[ x = 3 \][/tex]
### d) Finding the vertex:
The vertex form of a quadratic function [tex]\( f(x) = a(x-h)^2 + k \)[/tex] and the vertex [tex]\((h, k)\)[/tex] can be found using the point on the axis of symmetry.
We already found [tex]\( x = 3 \)[/tex] (the x-coordinate of the vertex). Now, substitute [tex]\( x = 3 \)[/tex] into the original function to find the y-coordinate:
[tex]\[ f(3) = 3(3)^2 - 18(3) + 24 \][/tex]
[tex]\[ f(3) = 3(9) - 54 + 24 \][/tex]
[tex]\[ f(3) = 27 - 54 + 24 \][/tex]
[tex]\[ f(3) = -3 \][/tex]
So, the vertex is:
[tex]\[ (3, -3) \][/tex]
### e) Sketching the graph of the function:
To sketch the graph of [tex]\( f(x) = 3x^2 - 18x + 24 \)[/tex]:
1. Identify and plot the y-intercept: [tex]\((0, 24)\)[/tex]
2. Identify and plot the x-intercepts: [tex]\((2, 0)\)[/tex] and [tex]\((4, 0)\)[/tex]
3. Identify and plot the vertex: [tex]\((3, -3)\)[/tex]
4. Draw the axis of symmetry: Vertical line through [tex]\(x = 3\)[/tex]
The graph is a parabola opening upwards (since the coefficient of [tex]\( x^2 \)[/tex] is positive) with the vertex as the lowest point.
### Graph:
```
|
25| X
20|
15|
10|
5|
0|----X--------X-------------------
2 3 4
-5| X Vertex(3, -3)
///(Parabola sketch)
```
Label key points:
- [tex]\( (0, 24) \)[/tex] as y-intercept
- [tex]\( (2, 0) \)[/tex] and [tex]\( (4, 0) \)[/tex] as x-intercepts
- [tex]\( (3, -3) \)[/tex] as vertex, with the axis of symmetry as [tex]\( x = 3 \)[/tex]
This careful plotting covers all necessary points and shapes required for an effective sketch.
