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Sagot :
To write [tex]\(q(x)\)[/tex] in vertex form and explain how it is transformed from the parent function [tex]\(p(x) = x^2\)[/tex], we start by expressing the quadratic function in vertex form.
### Vertex Form of a Quadratic Function
The vertex form of a quadratic function is given by:
[tex]\[ q(x) = a(x - h)^2 + k \][/tex]
where:
- [tex]\(a\)[/tex] controls the vertical stretch or compression and the direction of the parabola (opens upwards if [tex]\(a > 0\)[/tex], downwards if [tex]\(a < 0\)[/tex]).
- [tex]\((h, k)\)[/tex] is the vertex of the parabola, indicating the maximum or minimum point.
### Given Quadratic Function
Suppose the quadratic function we are considering is:
[tex]\[ q(x) = 2(x - 3)^2 + 5 \][/tex]
Here, we have:
- [tex]\(a = 2\)[/tex], which means the parabola opens upwards and is vertically stretched by a factor of 2.
- [tex]\(h = 3\)[/tex], which means the parabola is shifted 3 units to the right.
- [tex]\(k = 5\)[/tex], which means the parabola is shifted 5 units upwards.
### Expanding the Vertex Form to Standard Form
To convert the vertex form to the standard quadratic form [tex]\(ax^2 + bx + c\)[/tex], we can expand the expression:
[tex]\[ q(x) = 2(x - 3)^2 + 5 \][/tex]
First, expand the binomial:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
Then, multiply by the coefficient [tex]\(2\)[/tex]:
[tex]\[ 2(x^2 - 6x + 9) = 2x^2 - 12x + 18 \][/tex]
Finally, add the constant term [tex]\(5\)[/tex]:
[tex]\[ 2x^2 - 12x + 18 + 5 = 2x^2 - 12x + 23 \][/tex]
So, the quadratic function in standard form is:
[tex]\[ q(x) = 2x^2 - 12x + 23 \][/tex]
### Explanation of the Transformation
The parent function is the simplest quadratic function:
[tex]\[ p(x) = x^2 \][/tex]
The function [tex]\(q(x) = 2(x-3)^2 + 5\)[/tex] is a transformed version of the parent function [tex]\(p(x) = x^2\)[/tex]. Here's how it is transformed:
1. Vertical Stretch by a Factor of 2: The coefficient [tex]\(a = 2\)[/tex] stretches the graph vertically by a factor of 2. This means that for each unit increase in [tex]\(x\)[/tex], the value of [tex]\(q(x)\)[/tex] increases twice as fast compared to the parent function.
2. Horizontal Shift to the Right by 3 Units: The term [tex]\(x - 3\)[/tex] inside the squared term shifts the graph horizontally to the right by 3 units. The vertex of the parabola, which was originally at the origin [tex]\((0, 0)\)[/tex] for the parent function, is now at [tex]\((3, 0)\)[/tex].
3. Vertical Shift Upwards by 5 Units: The constant term [tex]\(+5\)[/tex] shifts the entire graph of the function upwards by 5 units. Therefore, the vertex that was at [tex]\((3, 0)\)[/tex] after the horizontal shift is moved up to [tex]\((3, 5)\)[/tex].
### Visual Transformation
- Starting with [tex]\(p(x) = x^2\)[/tex], the basic parabola opens upwards with its vertex at the origin.
- Stretching vertically by a factor of 2 makes it narrower.
- Shifting 3 units to the right moves the vertex to [tex]\( (3, 0) \)[/tex].
- Finally, shifting 5 units up moves the vertex to [tex]\( (3, 5) \)[/tex].
Therefore, the given function [tex]\( q(x) = 2(x - 3)^2 + 5 \)[/tex] is a vertically stretched, right-shifted, and upwards-shifted version of the parent function [tex]\( p(x) = x^2 \)[/tex]. The vertex form [tex]\( q(x) \)[/tex] effectively captures all the transformations applied to the original parent function.
### Vertex Form of a Quadratic Function
The vertex form of a quadratic function is given by:
[tex]\[ q(x) = a(x - h)^2 + k \][/tex]
where:
- [tex]\(a\)[/tex] controls the vertical stretch or compression and the direction of the parabola (opens upwards if [tex]\(a > 0\)[/tex], downwards if [tex]\(a < 0\)[/tex]).
- [tex]\((h, k)\)[/tex] is the vertex of the parabola, indicating the maximum or minimum point.
### Given Quadratic Function
Suppose the quadratic function we are considering is:
[tex]\[ q(x) = 2(x - 3)^2 + 5 \][/tex]
Here, we have:
- [tex]\(a = 2\)[/tex], which means the parabola opens upwards and is vertically stretched by a factor of 2.
- [tex]\(h = 3\)[/tex], which means the parabola is shifted 3 units to the right.
- [tex]\(k = 5\)[/tex], which means the parabola is shifted 5 units upwards.
### Expanding the Vertex Form to Standard Form
To convert the vertex form to the standard quadratic form [tex]\(ax^2 + bx + c\)[/tex], we can expand the expression:
[tex]\[ q(x) = 2(x - 3)^2 + 5 \][/tex]
First, expand the binomial:
[tex]\[ (x - 3)^2 = x^2 - 6x + 9 \][/tex]
Then, multiply by the coefficient [tex]\(2\)[/tex]:
[tex]\[ 2(x^2 - 6x + 9) = 2x^2 - 12x + 18 \][/tex]
Finally, add the constant term [tex]\(5\)[/tex]:
[tex]\[ 2x^2 - 12x + 18 + 5 = 2x^2 - 12x + 23 \][/tex]
So, the quadratic function in standard form is:
[tex]\[ q(x) = 2x^2 - 12x + 23 \][/tex]
### Explanation of the Transformation
The parent function is the simplest quadratic function:
[tex]\[ p(x) = x^2 \][/tex]
The function [tex]\(q(x) = 2(x-3)^2 + 5\)[/tex] is a transformed version of the parent function [tex]\(p(x) = x^2\)[/tex]. Here's how it is transformed:
1. Vertical Stretch by a Factor of 2: The coefficient [tex]\(a = 2\)[/tex] stretches the graph vertically by a factor of 2. This means that for each unit increase in [tex]\(x\)[/tex], the value of [tex]\(q(x)\)[/tex] increases twice as fast compared to the parent function.
2. Horizontal Shift to the Right by 3 Units: The term [tex]\(x - 3\)[/tex] inside the squared term shifts the graph horizontally to the right by 3 units. The vertex of the parabola, which was originally at the origin [tex]\((0, 0)\)[/tex] for the parent function, is now at [tex]\((3, 0)\)[/tex].
3. Vertical Shift Upwards by 5 Units: The constant term [tex]\(+5\)[/tex] shifts the entire graph of the function upwards by 5 units. Therefore, the vertex that was at [tex]\((3, 0)\)[/tex] after the horizontal shift is moved up to [tex]\((3, 5)\)[/tex].
### Visual Transformation
- Starting with [tex]\(p(x) = x^2\)[/tex], the basic parabola opens upwards with its vertex at the origin.
- Stretching vertically by a factor of 2 makes it narrower.
- Shifting 3 units to the right moves the vertex to [tex]\( (3, 0) \)[/tex].
- Finally, shifting 5 units up moves the vertex to [tex]\( (3, 5) \)[/tex].
Therefore, the given function [tex]\( q(x) = 2(x - 3)^2 + 5 \)[/tex] is a vertically stretched, right-shifted, and upwards-shifted version of the parent function [tex]\( p(x) = x^2 \)[/tex]. The vertex form [tex]\( q(x) \)[/tex] effectively captures all the transformations applied to the original parent function.
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