Question
Determine the zeros of the quadratic function y=x^2+15x+36 . (When listing the smaller zero, remember that negative values are always smaller than positive values.)
Answers
Answer
Determine the zeros of the quadratic function y=−x^2+16x−39 . (When listing the smaller zero, remember that negative values are always smaller than positive values.)
Answer
Determine the vertex and axis of symmetry of y=x^2−10x+3 .
Answer
What point is symmetrical to (−1,−3) for the graph of y=x^2+12x+8 ?
Answer
Question What are the key points on the graph of y=x^2−16x+48 ? Name the vertex, x-intercepts, and y-intercept.
Answered by
GPT 3.5
We can find the zeros of the quadratic function by setting y equal to zero and solving for x.
x^2 + 15x + 36 = 0
We can use factoring or the quadratic formula to solve for x.
Factoring:
(x + 3)(x + 12) = 0
Setting each factor equal to zero:
x + 3 = 0 or x + 12 = 0
x = -3 or x = -12
Therefore, the zeros of the quadratic function are x = -3 and x = -12. The smaller zero is x = -12, as negative values are always smaller than positive values.
x^2 + 15x + 36 = 0
We can use factoring or the quadratic formula to solve for x.
Factoring:
(x + 3)(x + 12) = 0
Setting each factor equal to zero:
x + 3 = 0 or x + 12 = 0
x = -3 or x = -12
Therefore, the zeros of the quadratic function are x = -3 and x = -12. The smaller zero is x = -12, as negative values are always smaller than positive values.
Answered by
GPT 3.5
To find the zeros of the quadratic function, we set y equal to zero and solve for x:
−x^2 + 16x - 39 = 0
We can use factoring or the quadratic formula to solve for x.
Factoring:
(x - 3)(x - 13) = 0
Setting each factor equal to zero:
x - 3 = 0 or x - 13 = 0
x = 3 or x = 13
Therefore, the zeros of the quadratic function are x = 3 and x = 13. The smaller zero is x = 3.
−x^2 + 16x - 39 = 0
We can use factoring or the quadratic formula to solve for x.
Factoring:
(x - 3)(x - 13) = 0
Setting each factor equal to zero:
x - 3 = 0 or x - 13 = 0
x = 3 or x = 13
Therefore, the zeros of the quadratic function are x = 3 and x = 13. The smaller zero is x = 3.
Answered by
GPT 3.5
To find the vertex and axis of symmetry of the quadratic function, we need to convert the given equation into vertex form, which is in the form y = a(x - h)^2 + k.
For the given equation y = x^2 - 10x + 3, we can complete the square to rewrite it into vertex form:
y = (x^2 - 10x) + 3
= (x^2 - 10x + 25) - 25 + 3
= (x - 5)^2 - 22
Comparing this with the vertex form, we can determine the vertex and axis of symmetry based on the values of h and k.
From the equation (x - 5)^2 + K (the vertex form), we can see that the vertex is at the point (h, k), where h = 5 and k = -22. So the vertex is (5, -22).
The axis of symmetry is a vertical line that passes through the vertex. In this case, since the x-coordinate of the vertex is 5, the equation of the axis of symmetry is x = 5.
For the given equation y = x^2 - 10x + 3, we can complete the square to rewrite it into vertex form:
y = (x^2 - 10x) + 3
= (x^2 - 10x + 25) - 25 + 3
= (x - 5)^2 - 22
Comparing this with the vertex form, we can determine the vertex and axis of symmetry based on the values of h and k.
From the equation (x - 5)^2 + K (the vertex form), we can see that the vertex is at the point (h, k), where h = 5 and k = -22. So the vertex is (5, -22).
The axis of symmetry is a vertical line that passes through the vertex. In this case, since the x-coordinate of the vertex is 5, the equation of the axis of symmetry is x = 5.
Answered by
GPT 3.5
To find the point that is symmetrical to (-1, -3) for the graph of y = x^2 + 12x + 8, we need to find the equation of the axis of symmetry and then reflect the given point across that axis.
The axis of symmetry of a quadratic function in the form y = ax^2 + bx + c is given by the equation x = -b/2a.
In this case, the equation is y = x^2 + 12x + 8, so a = 1 and b = 12. Plugging these values into the equation for the axis of symmetry gives:
x = -12 / (2 * 1)
x = -12/2
x = -6
The axis of symmetry is x = -6.
To find the point symmetrical to (-1, -3), we need to reflect it across the axis of symmetry.
The distance between the given point (-1, -3) and the axis of symmetry x = -6 is the same as the distance between the unknown point and x = -6.
The x-coordinate of the unknown point should be on the other side of x = -6, so it will have a positive x value.
Using the difference in x-coordinates:
Difference in x = -1 - (-6) = 5
The x-coordinate of the unknown point will be 5 units to the right of x = -6:
x = -6 + 5
x = -1
Since the axis of symmetry is a vertical line, the y-coordinate of the unknown point will be the same as the y-coordinate of the given point:
y = -3
Therefore, the point that is symmetrical to (-1, -3) for the graph of y = x^2 + 12x + 8 is (-1, -3).
The axis of symmetry of a quadratic function in the form y = ax^2 + bx + c is given by the equation x = -b/2a.
In this case, the equation is y = x^2 + 12x + 8, so a = 1 and b = 12. Plugging these values into the equation for the axis of symmetry gives:
x = -12 / (2 * 1)
x = -12/2
x = -6
The axis of symmetry is x = -6.
To find the point symmetrical to (-1, -3), we need to reflect it across the axis of symmetry.
The distance between the given point (-1, -3) and the axis of symmetry x = -6 is the same as the distance between the unknown point and x = -6.
The x-coordinate of the unknown point should be on the other side of x = -6, so it will have a positive x value.
Using the difference in x-coordinates:
Difference in x = -1 - (-6) = 5
The x-coordinate of the unknown point will be 5 units to the right of x = -6:
x = -6 + 5
x = -1
Since the axis of symmetry is a vertical line, the y-coordinate of the unknown point will be the same as the y-coordinate of the given point:
y = -3
Therefore, the point that is symmetrical to (-1, -3) for the graph of y = x^2 + 12x + 8 is (-1, -3).
Answered by
GPT 3.5
To find the key points on the graph of y = x^2 - 16x + 48, we can determine the vertex, x-intercepts, and y-intercept.
First, let's find the vertex. The vertex of a quadratic function in the form y = ax^2 + bx + c can be found using the formula x = -b/(2a). In this case, a = 1 and b = -16. Plugging these values into the formula gives:
x = -(-16) / (2 * 1)
x = 16 / 2
x = 8
To find the y-coordinate of the vertex, we substitute x = 8 into the equation:
y = (8)^2 - 16(8) + 48
y = 64 - 128 + 48
y = -16
So, the vertex is (8, -16).
Next, let's find the x-intercepts. These are the points where the graph intersects the x-axis. To find them, we set y = 0 and solve for x.
0 = x^2 - 16x + 48
This quadratic equation does not factor easily, so we can use the quadratic formula to find the x-intercepts:
x = (-(-16) ± sqrt((-16)^2 - 4(1)(48))) / (2(1))
x = (16 ± sqrt(256 - 192)) / 2
x = (16 ± sqrt(64)) / 2
x = (16 ± 8) / 2
Simplifying further, we get:
x = 8 ± 4
So, the x-intercepts are x = 12 and x = 4.
Lastly, let's find the y-intercept. This is the point where the graph intersects the y-axis. To find it, we set x = 0 and solve for y:
y = (0)^2 - 16(0) + 48
y = 48
So, the y-intercept is (0, 48).
In summary, the key points on the graph of y = x^2 - 16x + 48 are:
- Vertex: (8, -16)
- X-intercepts: (12, 0) and (4, 0)
- Y-intercept: (0, 48)
First, let's find the vertex. The vertex of a quadratic function in the form y = ax^2 + bx + c can be found using the formula x = -b/(2a). In this case, a = 1 and b = -16. Plugging these values into the formula gives:
x = -(-16) / (2 * 1)
x = 16 / 2
x = 8
To find the y-coordinate of the vertex, we substitute x = 8 into the equation:
y = (8)^2 - 16(8) + 48
y = 64 - 128 + 48
y = -16
So, the vertex is (8, -16).
Next, let's find the x-intercepts. These are the points where the graph intersects the x-axis. To find them, we set y = 0 and solve for x.
0 = x^2 - 16x + 48
This quadratic equation does not factor easily, so we can use the quadratic formula to find the x-intercepts:
x = (-(-16) ± sqrt((-16)^2 - 4(1)(48))) / (2(1))
x = (16 ± sqrt(256 - 192)) / 2
x = (16 ± sqrt(64)) / 2
x = (16 ± 8) / 2
Simplifying further, we get:
x = 8 ± 4
So, the x-intercepts are x = 12 and x = 4.
Lastly, let's find the y-intercept. This is the point where the graph intersects the y-axis. To find it, we set x = 0 and solve for y:
y = (0)^2 - 16(0) + 48
y = 48
So, the y-intercept is (0, 48).
In summary, the key points on the graph of y = x^2 - 16x + 48 are:
- Vertex: (8, -16)
- X-intercepts: (12, 0) and (4, 0)
- Y-intercept: (0, 48)