Use the image to answer the question. Identify the vertex on the graph. (1 point) Responses (5,0) left parenthesis 5 comma 0 right parenthesis (0,−5) left parenthesis 0 comma negative 5 right parenthesis (−5,0) left parenthesis negative 5 comma 0 right parenthesis x=−5 x equals negative 5
8 days ago
8 days ago
Determine the zero(s) of the quadratic function y=−5x2+16x.(1 point)
Responses
(0,5)(0,16)
left parenthesis 0 comma 5 right parenthesis left parenthesis 0 comma 16 right parenthesis
There are no zeros.
There are no zeros.
(−3.2,0)
left parenthesis negative 3.2 comma 0 right parenthesis
(0,0)(3.2,0)
left parenthesis 0 comma 0 right parenthesis left parenthesis 3.2 comma 0 right parenthesis
8 days ago
To determine the zeros of the quadratic function \( y = -5x^2 + 16x \), we need to set the equation equal to zero and solve for \( x \):
\[
-5x^2 + 16x = 0
\]
Factor out \( x \):
\[
x(-5x + 16) = 0
\]
This gives us two factors to set equal to zero:
1. \( x = 0 \)
2. \( -5x + 16 = 0 \)
Now, solve the second equation:
\[
-5x + 16 = 0 \implies 5x = 16 \implies x = \frac{16}{5} = 3.2
\]
Thus, the zeros of the function are \( x = 0 \) and \( x = 3.2 \). The corresponding points on the graph are \( (0, 0) \) and \( (3.2, 0) \).
**The correct response is: (0,0) (3.2,0).**
8 days ago
Identify the vertex of y=4(x−25)2−61.(1 point)
Responses
(−25,−61)
left parenthesis negative 25 comma negative 61 right parenthesis
(−100,−61)
left parenthesis negative 100 comma negative 61 right parenthesis
(4,−25)
left parenthesis 4 comma negative 25 right parenthesis
(25,−61)
left parenthesis 25 comma negative 61 right parenthesis
8 days ago
The given equation is in vertex form:
\[
y = a(x - h)^2 + k
\]
where \((h, k)\) is the vertex. In this case, the equation is:
\[
y = 4(x - 25)^2 - 61
\]
From this, we can see:
- \(h = 25\)
- \(k = -61\)
Thus, the vertex is \((25, -61)\).
**The correct response is: (25, -61).**
8 days ago
What are the key points on the graph of y=x2−2x−120? Name the vertex, x-intercept(s), and y-intercept.(1 point)
Responses
x-intercept: (1,−121)
y-intercept: (0,−120)
vertex: (−10,0) (12,0)
x -intercept: left parenthesis 1 comma negative 121 right parenthesis y -intercept: left parenthesis 0 comma negative 120 right parenthesis vertex: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis
x-intercepts: (−10,0) (12,0)
y-intercept: (0,−120)
vertex: (1,−121)
x -intercepts: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis y -intercept: left parenthesis 0 comma negative 120 right parenthesis vertex: left parenthesis 1 comma negative 121 right parenthesis
x-intercepts: (−10,0) (12,0)
y-intercept: (1,−121)
vertex: (0,−120)
x -intercepts: left parenthesis negative 10 comma 0 right parenthesis left parenthesis 12 comma 0 right parenthesis y-intercept: left parenthesis 1 comma negative 121 right parenthesis vertex: left parenthesis 0 comma negative 120 right parenthesis
x-intercept: (0,−120)
y-intercept: (−10,0) (12,0)
vertex: (1,−121)
8 days ago
To find the key points on the graph of the quadratic function \( y = x^2 - 2x - 120 \), we can:
1. **Find the vertex** using the formula \( x = -\frac{b}{2a} \), where \( a = 1 \) and \( b = -2 \) in the general form \( y = ax^2 + bx + c \).
\[
x = -\frac{-2}{2(1)} = \frac{2}{2} = 1
\]
Now, substitute \( x = 1 \) back into the equation to find \( y \):
\[
y = (1)^2 - 2(1) - 120 = 1 - 2 - 120 = -121
\]
So the vertex is \( (1, -121) \).
2. **Find the x-intercepts** by setting \( y = 0 \):
\[
0 = x^2 - 2x - 120
\]
Factoring the equation:
\[
0 = (x - 12)(x + 10)
\]
Setting each factor to zero gives:
\[
x - 12 = 0 \implies x = 12 \quad \text{and} \quad x + 10 = 0 \implies x = -10
\]
Thus, the x-intercepts are \( (-10, 0) \) and \( (12, 0) \).
3. **Find the y-intercept** by setting \( x = 0 \):
\[
y = (0)^2 - 2(0) - 120 = -120
\]
Thus, the y-intercept is \( (0, -120) \).
### Summary of Key Points
- **Vertex**: \( (1, -121) \)
- **X-intercepts**: \( (-10, 0) \) and \( (12, 0) \)
- **Y-intercept**: \( (0, -120) \)
**The correct response is:**
- x-intercepts: \( (-10, 0) \) and \( (12, 0) \)
- y-intercept: \( (0, -120) \)
- vertex: \( (1, -121) \)