Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Experience the convenience of finding accurate answers to your questions from knowledgeable professionals on our platform. Our platform provides a seamless experience for finding reliable answers from a network of experienced professionals.
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]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Get the answers you need at Westonci.ca. Stay informed by returning for our latest expert advice.
