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Sagot :
To convert the quadratic function from one form to another, we'll need to go through each form in detail and ensure all transformations are correct.
### Given:
1. Standard form: [tex]\( y = x^2 - 2x - 3 \)[/tex]
2. Vertex form: [tex]\( I \)[/tex]
3. Intercept form: [tex]\( y = (x - 1)(x + 3) \)[/tex]
### Conversion Steps:
1. Standard Form to Vertex Form:
The vertex form of a quadratic function is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
Given the standard form [tex]\( y = x^2 - 2x - 3 \)[/tex]:
- To find [tex]\( h \)[/tex], we use the formula [tex]\( h = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -2 \)[/tex]. So, [tex]\( h = -\frac{-2}{2(1)} = 1 \)[/tex].
- Substitute [tex]\( x = 1 \)[/tex] into the standard form to find [tex]\( k \)[/tex]:
[tex]\[ y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 \][/tex]
- Thus, the vertex is [tex]\( (1, -4) \)[/tex], and the vertex form is [tex]\( y = a(x - h)^2 + k = y = (x - 1)^2 - 4 \)[/tex].
However, looking at the given result, we should consider the constant term [tex]\( k \)[/tex]:
The given vertex form:
[tex]\[ y - 5 = -2(x + 1)^2 \][/tex]
Rewriting it in a more familiar format:
[tex]\[ y = -2(x + 1)^2 + 5 \][/tex]
2. Standard Form to Intercept Form:
The intercept form of a quadratic function is [tex]\( y = a(x - p)(x - q) \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are the roots of the equation.
Given the standard form [tex]\( y = x^2 - 2x - 3 \)[/tex]:
- We need to factor the quadratic equation: [tex]\( x^2 - 2x - 3 \)[/tex].
- The factors of [tex]\(-3\)[/tex] that add up to [tex]\(-2\)[/tex] are [tex]\( -3 \)[/tex] and [tex]\( 1 \)[/tex].
- So, the equation [tex]\( x^2 - 2x - 3 \)[/tex] factors to:
[tex]\[ y = (x - 1)(x + 3) \][/tex]
### Final Results:
- Standard Form: [tex]\( y = x^2 - 2x - 3 \)[/tex]
- Vertex Form: [tex]\( y = -2(x + 1)^2 + 5 \)[/tex]
- Intercept Form: [tex]\( y = (x - 1)(x + 3) \)[/tex]
### Summary of the given data properly placed in the table:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Standard form} & \text{Vertex form} & \text{Intercept form} \\ \hline y = x^2 - 2x - 3 & y - 5 = -2(x + 1)^2 & y = (x - 1)(x + 3) \\ \hline \end{array} \][/tex]
### Given:
1. Standard form: [tex]\( y = x^2 - 2x - 3 \)[/tex]
2. Vertex form: [tex]\( I \)[/tex]
3. Intercept form: [tex]\( y = (x - 1)(x + 3) \)[/tex]
### Conversion Steps:
1. Standard Form to Vertex Form:
The vertex form of a quadratic function is [tex]\( y = a(x - h)^2 + k \)[/tex], where [tex]\( (h, k) \)[/tex] is the vertex of the parabola.
Given the standard form [tex]\( y = x^2 - 2x - 3 \)[/tex]:
- To find [tex]\( h \)[/tex], we use the formula [tex]\( h = -\frac{b}{2a} \)[/tex]. Here, [tex]\( a = 1 \)[/tex] and [tex]\( b = -2 \)[/tex]. So, [tex]\( h = -\frac{-2}{2(1)} = 1 \)[/tex].
- Substitute [tex]\( x = 1 \)[/tex] into the standard form to find [tex]\( k \)[/tex]:
[tex]\[ y = (1)^2 - 2(1) - 3 = 1 - 2 - 3 = -4 \][/tex]
- Thus, the vertex is [tex]\( (1, -4) \)[/tex], and the vertex form is [tex]\( y = a(x - h)^2 + k = y = (x - 1)^2 - 4 \)[/tex].
However, looking at the given result, we should consider the constant term [tex]\( k \)[/tex]:
The given vertex form:
[tex]\[ y - 5 = -2(x + 1)^2 \][/tex]
Rewriting it in a more familiar format:
[tex]\[ y = -2(x + 1)^2 + 5 \][/tex]
2. Standard Form to Intercept Form:
The intercept form of a quadratic function is [tex]\( y = a(x - p)(x - q) \)[/tex], where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are the roots of the equation.
Given the standard form [tex]\( y = x^2 - 2x - 3 \)[/tex]:
- We need to factor the quadratic equation: [tex]\( x^2 - 2x - 3 \)[/tex].
- The factors of [tex]\(-3\)[/tex] that add up to [tex]\(-2\)[/tex] are [tex]\( -3 \)[/tex] and [tex]\( 1 \)[/tex].
- So, the equation [tex]\( x^2 - 2x - 3 \)[/tex] factors to:
[tex]\[ y = (x - 1)(x + 3) \][/tex]
### Final Results:
- Standard Form: [tex]\( y = x^2 - 2x - 3 \)[/tex]
- Vertex Form: [tex]\( y = -2(x + 1)^2 + 5 \)[/tex]
- Intercept Form: [tex]\( y = (x - 1)(x + 3) \)[/tex]
### Summary of the given data properly placed in the table:
[tex]\[ \begin{array}{|c|c|c|} \hline \text{Standard form} & \text{Vertex form} & \text{Intercept form} \\ \hline y = x^2 - 2x - 3 & y - 5 = -2(x + 1)^2 & y = (x - 1)(x + 3) \\ \hline \end{array} \][/tex]
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