Answer :
Let's go through this step by step, ensuring that we understand each part of the problem clearly:
### (a) Find the vertex and axis of symmetry of the quadratic function.
We are given the quadratic function:
[tex]\[ f(x) = (x - 6)^2 + 3 \][/tex]
This function is in the standard form of a quadratic function written as:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
From the given function [tex]\((x - 6)^2 + 3\)[/tex]:
- [tex]\((h, k)\)[/tex] = (6, 3)
Thus, the vertex of the quadratic function is:
[tex]\[ (6, 3) \][/tex]
The axis of symmetry of a quadratic function in this form is [tex]\( x = h \)[/tex]. Therefore, the axis of symmetry for this function is:
[tex]\[ x = 6 \][/tex]
### (b) Determine whether the graph is concave up or concave down.
To determine whether the graph is concave up or concave down, we need to examine the coefficient of the squared term, [tex]\(a\)[/tex], in the standard form of the quadratic equation [tex]\( f(x) = a(x - h)^2 + k \)[/tex].
In the given function:
[tex]\[ f(x) = (x - 6)^2 + 3 \][/tex]
The coefficient [tex]\(a\)[/tex] is 1 (which is positive). If [tex]\(a > 0\)[/tex], the graph is concave up. If [tex]\(a < 0\)[/tex], the graph is concave down.
Since [tex]\(a = 1\)[/tex] which is positive, the graph is:
[tex]\[ \text{concave up} \][/tex]
### (c) Graph the quadratic function.
To graph this quadratic function [tex]\( f(x) = (x - 6)^2 + 3 \)[/tex], we follow these steps:
1. Plot the Vertex:
The vertex of the function is [tex]\( (6, 3) \)[/tex]. So we plot the point (6, 3) on the coordinate plane.
2. Draw the Axis of Symmetry:
The axis of symmetry is the vertical line [tex]\( x = 6 \)[/tex]. This line divides the parabola into two mirror images.
3. Determine Additional Points:
Choose a couple of additional points to plot on each side of the vertex and reflect them across the axis of symmetry to get a better idea of the shape of the parabola.
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = (5 - 6)^2 + 3 = 1^2 + 3 = 4 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = (7 - 6)^2 + 3 = 1^2 + 3 = 4 \][/tex]
So, the points (5, 4) and (7, 4) are on the parabola.
4. Sketch the Parabola:
Using the vertex (6, 3), and the points (5, 4) and (7, 4), along with the symmetrical property around the axis of symmetry [tex]\(x = 6\)[/tex], sketch the parabola opening upwards.
### Final Summary:
- Vertex: (6, 3)
- Axis of Symmetry: [tex]\( x = 6 \)[/tex]
- Concavity: Concave up
- Graph: Sketch with the vertex, axis of symmetry, and a few calculated points reflecting the symmetrical nature around the axis.
By following these detailed steps, we can effectively graph and understand the quadratic function [tex]\( f(x) = (x-6)^2 + 3 \)[/tex].
### (a) Find the vertex and axis of symmetry of the quadratic function.
We are given the quadratic function:
[tex]\[ f(x) = (x - 6)^2 + 3 \][/tex]
This function is in the standard form of a quadratic function written as:
[tex]\[ f(x) = a(x - h)^2 + k \][/tex]
where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
From the given function [tex]\((x - 6)^2 + 3\)[/tex]:
- [tex]\((h, k)\)[/tex] = (6, 3)
Thus, the vertex of the quadratic function is:
[tex]\[ (6, 3) \][/tex]
The axis of symmetry of a quadratic function in this form is [tex]\( x = h \)[/tex]. Therefore, the axis of symmetry for this function is:
[tex]\[ x = 6 \][/tex]
### (b) Determine whether the graph is concave up or concave down.
To determine whether the graph is concave up or concave down, we need to examine the coefficient of the squared term, [tex]\(a\)[/tex], in the standard form of the quadratic equation [tex]\( f(x) = a(x - h)^2 + k \)[/tex].
In the given function:
[tex]\[ f(x) = (x - 6)^2 + 3 \][/tex]
The coefficient [tex]\(a\)[/tex] is 1 (which is positive). If [tex]\(a > 0\)[/tex], the graph is concave up. If [tex]\(a < 0\)[/tex], the graph is concave down.
Since [tex]\(a = 1\)[/tex] which is positive, the graph is:
[tex]\[ \text{concave up} \][/tex]
### (c) Graph the quadratic function.
To graph this quadratic function [tex]\( f(x) = (x - 6)^2 + 3 \)[/tex], we follow these steps:
1. Plot the Vertex:
The vertex of the function is [tex]\( (6, 3) \)[/tex]. So we plot the point (6, 3) on the coordinate plane.
2. Draw the Axis of Symmetry:
The axis of symmetry is the vertical line [tex]\( x = 6 \)[/tex]. This line divides the parabola into two mirror images.
3. Determine Additional Points:
Choose a couple of additional points to plot on each side of the vertex and reflect them across the axis of symmetry to get a better idea of the shape of the parabola.
- For [tex]\( x = 5 \)[/tex]:
[tex]\[ f(5) = (5 - 6)^2 + 3 = 1^2 + 3 = 4 \][/tex]
- For [tex]\( x = 7 \)[/tex]:
[tex]\[ f(7) = (7 - 6)^2 + 3 = 1^2 + 3 = 4 \][/tex]
So, the points (5, 4) and (7, 4) are on the parabola.
4. Sketch the Parabola:
Using the vertex (6, 3), and the points (5, 4) and (7, 4), along with the symmetrical property around the axis of symmetry [tex]\(x = 6\)[/tex], sketch the parabola opening upwards.
### Final Summary:
- Vertex: (6, 3)
- Axis of Symmetry: [tex]\( x = 6 \)[/tex]
- Concavity: Concave up
- Graph: Sketch with the vertex, axis of symmetry, and a few calculated points reflecting the symmetrical nature around the axis.
By following these detailed steps, we can effectively graph and understand the quadratic function [tex]\( f(x) = (x-6)^2 + 3 \)[/tex].
