Question
Graph the equation y=−5(x)2+1 . Which of the following statements about the graph is correct?(1 point)
Responses
The parabola will open downwards.
The parabola will open downwards.
The parabola will have a vertex at (1,0).
The parabola will have a vertex at left parenthesis 1 comma 0 right parenthesis .
The parabola will have a vertex at (−1,0).
The parabola will have a vertex at left parenthesis negative 1 comma 0 right parenthesis .
The parabola will have a vertex at (0,0).
The parabola will have a vertex at left parenthesis 0 comma 0 right parenthesis .
Responses
The parabola will open downwards.
The parabola will open downwards.
The parabola will have a vertex at (1,0).
The parabola will have a vertex at left parenthesis 1 comma 0 right parenthesis .
The parabola will have a vertex at (−1,0).
The parabola will have a vertex at left parenthesis negative 1 comma 0 right parenthesis .
The parabola will have a vertex at (0,0).
The parabola will have a vertex at left parenthesis 0 comma 0 right parenthesis .
Answers
There are no human answers yet.
Answered by
GPT-4o mini
To analyze the equation \( y = -5(x)^2 + 1 \), let's break it down:
1. **Graphing the Equation**: The equation is in the standard quadratic form \( y = ax^2 + bx + c \), where \( a = -5 \), \( b = 0 \), and \( c = 1 \).
- Since \( a < 0 \), the parabola opens downwards.
2. **Finding the Vertex**: The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( b = 0 \). Therefore, the x-coordinate of the vertex is:
\[
x = -\frac{0}{2(-5)} = 0
\]
- To find the y-coordinate of the vertex, we substitute \( x = 0 \) back into the equation:
\[
y = -5(0)^2 + 1 = 1
\]
- Thus, the vertex is at \( (0, 1) \).
3. **Statements Assessment**:
- The first statement, "The parabola will open downwards," is **correct**.
- The second statement, "The parabola will have a vertex at (1, 0)," is **incorrect**.
- The third statement, "The parabola will have a vertex at (−1, 0)," is **incorrect**.
- The fourth statement, "The parabola will have a vertex at (0, 0)," is also **incorrect**.
Based on the analysis, the correct answer is:
**The parabola will open downwards.**
1. **Graphing the Equation**: The equation is in the standard quadratic form \( y = ax^2 + bx + c \), where \( a = -5 \), \( b = 0 \), and \( c = 1 \).
- Since \( a < 0 \), the parabola opens downwards.
2. **Finding the Vertex**: The vertex of a parabola in the form \( y = ax^2 + bx + c \) can be found using the formula \( x = -\frac{b}{2a} \).
- Here, \( b = 0 \). Therefore, the x-coordinate of the vertex is:
\[
x = -\frac{0}{2(-5)} = 0
\]
- To find the y-coordinate of the vertex, we substitute \( x = 0 \) back into the equation:
\[
y = -5(0)^2 + 1 = 1
\]
- Thus, the vertex is at \( (0, 1) \).
3. **Statements Assessment**:
- The first statement, "The parabola will open downwards," is **correct**.
- The second statement, "The parabola will have a vertex at (1, 0)," is **incorrect**.
- The third statement, "The parabola will have a vertex at (−1, 0)," is **incorrect**.
- The fourth statement, "The parabola will have a vertex at (0, 0)," is also **incorrect**.
Based on the analysis, the correct answer is:
**The parabola will open downwards.**
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