Find solutions to your problems with the help of IDNLearn.com's knowledgeable users. Find in-depth and accurate answers to all your questions from our knowledgeable and dedicated community members.
Sagot :
Let's tackle the problem of graphing the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex].
### Step-by-Step Solution
1. Expand the Equation:
Rewrite the quadratic equation in the standard form.
[tex]\[ f(x) = (x-2)(x-6) \][/tex]
Expanding this, we get:
[tex]\[ f(x) = x^2 - 6x - 2x + 12 \][/tex]
[tex]\[ f(x) = x^2 - 8x + 12 \][/tex]
2. Identify the Vertex:
The vertex form of a quadratic equation is [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
For the given quadratic equation [tex]\( y = x^2 - 8x + 12 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 12 \)[/tex]
To find the [tex]\( x \)[/tex]-coordinate of the vertex [tex]\( h \)[/tex], use the formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Substituting the values:
[tex]\[ h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \][/tex]
Now, find the [tex]\( y \)[/tex]-coordinate [tex]\( k \)[/tex] by substituting [tex]\( h = 4 \)[/tex] back into the function:
[tex]\[ k = f(4) = (4-2)(4-6) = 2 \cdot -2 = -4 \][/tex]
So, the vertex of the parabola is [tex]\((4, -4)\)[/tex].
3. Choose Another Point:
Select another point on the parabola for accuracy. We chose [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = (3-2)(3-6) = 1 \cdot -3 = -3 \][/tex]
Therefore, another point on the parabola is [tex]\((3, -3)\)[/tex].
4. Plot the Points:
Use the vertex [tex]\((4, -4)\)[/tex] and the additional point [tex]\((3, -3)\)[/tex] to plot the quadratic function.
To graph [tex]\( f(x) \)[/tex]:
- Plot the vertex at [tex]\((4, -4)\)[/tex].
- Plot the additional point at [tex]\((3, -3)\)[/tex].
- Draw a symmetrical parabola passing through these points, opening upwards since the coefficient of [tex]\( x^2 \)[/tex] is positive (i.e., [tex]\( a = 1 \)[/tex]).
By following these steps, you can accurately graph the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex] using the vertex and another point on the parabola.
### Step-by-Step Solution
1. Expand the Equation:
Rewrite the quadratic equation in the standard form.
[tex]\[ f(x) = (x-2)(x-6) \][/tex]
Expanding this, we get:
[tex]\[ f(x) = x^2 - 6x - 2x + 12 \][/tex]
[tex]\[ f(x) = x^2 - 8x + 12 \][/tex]
2. Identify the Vertex:
The vertex form of a quadratic equation is [tex]\( y = a(x-h)^2 + k \)[/tex], where [tex]\((h, k)\)[/tex] is the vertex of the parabola.
For the given quadratic equation [tex]\( y = x^2 - 8x + 12 \)[/tex]:
- [tex]\( a = 1 \)[/tex]
- [tex]\( b = -8 \)[/tex]
- [tex]\( c = 12 \)[/tex]
To find the [tex]\( x \)[/tex]-coordinate of the vertex [tex]\( h \)[/tex], use the formula:
[tex]\[ h = -\frac{b}{2a} \][/tex]
Substituting the values:
[tex]\[ h = -\frac{-8}{2 \cdot 1} = \frac{8}{2} = 4 \][/tex]
Now, find the [tex]\( y \)[/tex]-coordinate [tex]\( k \)[/tex] by substituting [tex]\( h = 4 \)[/tex] back into the function:
[tex]\[ k = f(4) = (4-2)(4-6) = 2 \cdot -2 = -4 \][/tex]
So, the vertex of the parabola is [tex]\((4, -4)\)[/tex].
3. Choose Another Point:
Select another point on the parabola for accuracy. We chose [tex]\( x = 3 \)[/tex]:
[tex]\[ f(3) = (3-2)(3-6) = 1 \cdot -3 = -3 \][/tex]
Therefore, another point on the parabola is [tex]\((3, -3)\)[/tex].
4. Plot the Points:
Use the vertex [tex]\((4, -4)\)[/tex] and the additional point [tex]\((3, -3)\)[/tex] to plot the quadratic function.
To graph [tex]\( f(x) \)[/tex]:
- Plot the vertex at [tex]\((4, -4)\)[/tex].
- Plot the additional point at [tex]\((3, -3)\)[/tex].
- Draw a symmetrical parabola passing through these points, opening upwards since the coefficient of [tex]\( x^2 \)[/tex] is positive (i.e., [tex]\( a = 1 \)[/tex]).
By following these steps, you can accurately graph the quadratic function [tex]\( f(x) = (x-2)(x-6) \)[/tex] using the vertex and another point on the parabola.
We value your participation in this forum. Keep exploring, asking questions, and sharing your insights with the community. Together, we can find the best solutions. Your search for solutions ends here at IDNLearn.com. Thank you for visiting, and come back soon for more helpful information.