Explore a diverse range of topics and get answers from knowledgeable individuals on IDNLearn.com. Our platform is designed to provide accurate and comprehensive answers to any questions you may have.
Sagot :
To determine which graph represents the equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex], let's go through a step-by-step analysis.
### Step 1: Understand the Form of the Equation
The given equation is a quadratic equation in the form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\( a = -1 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 1 \)[/tex].
### Step 2: Identify the Vertex
The vertex of a parabola given by the equation [tex]\( y = a(x - h)^2 + k \)[/tex] is located at the point [tex]\( (h, k) \)[/tex].
Therefore, the vertex of the given equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is at:
[tex]\[ (h, k) = (1, 1) \][/tex]
### Step 3: Determine the Direction of the Parabola
The coefficient [tex]\( a \)[/tex] in front of the squared term determines the direction in which the parabola opens:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
Since [tex]\( a = -1 \)[/tex], which is less than 0, this parabola opens downwards.
### Step 4: Sketch and Analyze the Parabola
Based on the vertex and the direction, let's sketch the parabola:
- The vertex is at [tex]\( (1, 1) \)[/tex].
- The parabola opens downwards.
The general shape of this parabola would look like an inverted "U" shape centered at the vertex.
### Step 5: Confirm the Shape by Plugging in Values (Optional Check)
For additional confirmation, you can plug in a few values of [tex]\( x \)[/tex] to find corresponding values of [tex]\( y \)[/tex] and plot those points:
1. When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1 - 1)^2 + 1 = 1 \][/tex]
Point: [tex]\( (1, 1) \)[/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (0, 0) \)[/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (2, 0) \)[/tex]
4. When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1 - 1)^2 + 1 = -4 + 1 = -3 \][/tex]
Point: [tex]\( (-1, -3) \)[/tex]
These points should lie on the parabola, and you can connect them to visualize its shape.
### Conclusion
The graph representing the equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is a downward-opening parabola with its vertex located at [tex]\( (1, 1) \)[/tex]. It is symmetric around the vertical line [tex]\( x = 1 \)[/tex] and will intersect the [tex]\( y \)[/tex]-axis somewhere below the vertex.
### Step 1: Understand the Form of the Equation
The given equation is a quadratic equation in the form:
[tex]\[ y = a(x - h)^2 + k \][/tex]
where [tex]\( a = -1 \)[/tex], [tex]\( h = 1 \)[/tex], and [tex]\( k = 1 \)[/tex].
### Step 2: Identify the Vertex
The vertex of a parabola given by the equation [tex]\( y = a(x - h)^2 + k \)[/tex] is located at the point [tex]\( (h, k) \)[/tex].
Therefore, the vertex of the given equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is at:
[tex]\[ (h, k) = (1, 1) \][/tex]
### Step 3: Determine the Direction of the Parabola
The coefficient [tex]\( a \)[/tex] in front of the squared term determines the direction in which the parabola opens:
- If [tex]\( a > 0 \)[/tex], the parabola opens upwards.
- If [tex]\( a < 0 \)[/tex], the parabola opens downwards.
Since [tex]\( a = -1 \)[/tex], which is less than 0, this parabola opens downwards.
### Step 4: Sketch and Analyze the Parabola
Based on the vertex and the direction, let's sketch the parabola:
- The vertex is at [tex]\( (1, 1) \)[/tex].
- The parabola opens downwards.
The general shape of this parabola would look like an inverted "U" shape centered at the vertex.
### Step 5: Confirm the Shape by Plugging in Values (Optional Check)
For additional confirmation, you can plug in a few values of [tex]\( x \)[/tex] to find corresponding values of [tex]\( y \)[/tex] and plot those points:
1. When [tex]\( x = 1 \)[/tex]:
[tex]\[ y = -(1 - 1)^2 + 1 = 1 \][/tex]
Point: [tex]\( (1, 1) \)[/tex]
2. When [tex]\( x = 0 \)[/tex]:
[tex]\[ y = -(0 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (0, 0) \)[/tex]
3. When [tex]\( x = 2 \)[/tex]:
[tex]\[ y = -(2 - 1)^2 + 1 = -(1) + 1 = 0 \][/tex]
Point: [tex]\( (2, 0) \)[/tex]
4. When [tex]\( x = -1 \)[/tex]:
[tex]\[ y = -(-1 - 1)^2 + 1 = -4 + 1 = -3 \][/tex]
Point: [tex]\( (-1, -3) \)[/tex]
These points should lie on the parabola, and you can connect them to visualize its shape.
### Conclusion
The graph representing the equation [tex]\( y = -(x - 1)^2 + 1 \)[/tex] is a downward-opening parabola with its vertex located at [tex]\( (1, 1) \)[/tex]. It is symmetric around the vertical line [tex]\( x = 1 \)[/tex] and will intersect the [tex]\( y \)[/tex]-axis somewhere below the vertex.
We value your presence here. Keep sharing knowledge and helping others find the answers they need. This community is the perfect place to learn together. For trustworthy answers, rely on IDNLearn.com. Thanks for visiting, and we look forward to assisting you again.
