Question
What is the vertex form of the equation graphed below?
graph shows a line that curves at 1,3
(1 point)
Responses
y=3(x+1)2−3
y is equal to 3 times open paren x plus 1 close paren squared minus 3
y=3(x−1)2+3
y is equal to 3 times open paren x minus 1 close paren squared plus 3
y=−2(x+1)2−3
y is equal to negative 2 times open paren x plus 1 close paren squared minus 3
y=−2(x−1)2+3
graph shows a line that curves at 1,3
(1 point)
Responses
y=3(x+1)2−3
y is equal to 3 times open paren x plus 1 close paren squared minus 3
y=3(x−1)2+3
y is equal to 3 times open paren x minus 1 close paren squared plus 3
y=−2(x+1)2−3
y is equal to negative 2 times open paren x plus 1 close paren squared minus 3
y=−2(x−1)2+3
Answers
GPT-4o mini
To find the vertex form of the equation graphed, we need to identify the vertex of the parabola, which is given as the point (1, 3).
The vertex form of a quadratic equation is given by:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola, and \( a \) determines the direction and width of the parabola.
From the provided options, we have the vertex at (1, 3), which means:
- \( h = 1 \)
- \( k = 3 \)
This gives us the base of our equation:
\[ y = a(x - 1)^2 + 3 \]
Now we need to determine the value of \( a \). The problem suggests the parabola opens downwards because the vertex is at a maximum point (3, which is the highest point). Hence, \( a \) should be negative.
The options provided are:
1. \( y = 3(x + 1)^2 - 3 \)
2. \( y = 3(x - 1)^2 + 3 \)
3. \( y = -2(x + 1)^2 - 3 \)
4. \( y = -2(x - 1)^2 + 3 \)
The only option that matches the vertex and has a negative value for \( a \) is:
**Option 4: \( y = -2(x - 1)^2 + 3 \)**
Thus, the correct answer is:
**y = -2(x - 1)² + 3**
The vertex form of a quadratic equation is given by:
\[ y = a(x - h)^2 + k \]
where \( (h, k) \) is the vertex of the parabola, and \( a \) determines the direction and width of the parabola.
From the provided options, we have the vertex at (1, 3), which means:
- \( h = 1 \)
- \( k = 3 \)
This gives us the base of our equation:
\[ y = a(x - 1)^2 + 3 \]
Now we need to determine the value of \( a \). The problem suggests the parabola opens downwards because the vertex is at a maximum point (3, which is the highest point). Hence, \( a \) should be negative.
The options provided are:
1. \( y = 3(x + 1)^2 - 3 \)
2. \( y = 3(x - 1)^2 + 3 \)
3. \( y = -2(x + 1)^2 - 3 \)
4. \( y = -2(x - 1)^2 + 3 \)
The only option that matches the vertex and has a negative value for \( a \) is:
**Option 4: \( y = -2(x - 1)^2 + 3 \)**
Thus, the correct answer is:
**y = -2(x - 1)² + 3**