Identify the vertex and axis of symetry of the parabola. Identify points corresponding to P and Q. on the graph it has P at (-2,4) and a point at Q at (-1,7). my teacher says the answers were the vertex was (-1,3) the Axis of symmtry was X=-1 the P is (0,4) and Q is (-3,7).. how do you figure out each of those steps? im confused with how to do the whole problem
I will try to reverse engineer this mess.
(y-3) = k (x+1)^2
if the vertex is at (-1,3) and x=-1 is the axis
put in P at (-1,7) impossible because y = 3 when x = -1
so I will try the second pair of points
P at (0,4)
(4-3) = k (1)^2
so k = 1
so
y - 3 = (x+1)^2
try Q at (-3,7)
4 = (-2)^2
YES
so the parabola with axis of symmetry at x = -1 and vertex at (-1,3) can contain the SECOND pair of points P and Q
rewrite in standard form
y = 3 +x^2 + 2 x + 1
y = x^2 + 2 x + 4
I have no idea what the two original points P and Q were. They are on some other curve.
The P and Q coordinates in your first statement do not agree with the P and Q in your last statement. I would need to see the graph to know what this problem is all about.
To identify the vertex and axis of symmetry of a parabola, you can use the vertex formula, which is -(b/2a) for a quadratic equation in the form of y = ax^2 + bx + c.
Let's take a look at the given information step by step to find the correct answers.
1. Given the point P at (-2,4): The x-coordinate of the vertex is the same as the x-coordinate of P. Therefore, the x-coordinate of the vertex is -2.
2. Given the point Q at (-1,7): Similar to the previous step, the x-coordinate of the vertex is the same as the x-coordinate of Q. Therefore, the x-coordinate of the vertex is -1.
3. Given the vertex as (-1,3): Since the x-coordinate of the vertex is -1 and we are given the point Q as (-1,7), we can see that the y-coordinate of the vertex is 7 - 4 = 3. Therefore, the vertex is (-1,3).
4. To find the axis of symmetry, we use the x-coordinate of the vertex. So, the axis of symmetry is x = -1.
5. To find the points corresponding to P and Q on the graph:
- For point P: Since the x-coordinate of P is -2 and the y-coordinate is 4, we can conclude that the y-coordinate of the point corresponding to P on the graph would be the same (4), and the x-coordinate would be the x-coordinate of the vertex plus the x-coordinate of P, which is -1 + (-2) = -3. Therefore, the point corresponding to P on the graph is (-3,4).
- For point Q: Since the x-coordinate of Q is -1 and the y-coordinate is 7, we can conclude that the y-coordinate of the point corresponding to Q on the graph would be the same (7), and the x-coordinate would be the x-coordinate of the vertex plus the x-coordinate of Q, which is -1 + (-1) = -2. Therefore, the point corresponding to Q on the graph is (-2,7).
Hence, the correct answers are:
- Vertex: (-1, 3)
- Axis of symmetry: X = -1
- Point P on the graph: (-3, 4)
- Point Q on the graph: (-2, 7)
To find the vertex and axis of symmetry of a parabola, you need to understand the concept of a quadratic function and analyze its equation.
1. Quadratic Function:
A quadratic function is a function that can be written in the form f(x) = ax^2 + bx + c, where a, b, and c are constants. In the case of a parabola, the coefficient "a" determines the direction and width of the parabola. If "a" is positive, the parabola opens upwards, and if "a" is negative, the parabola opens downwards.
2. Vertex Form of a Quadratic Function:
The vertex form of a quadratic function is given by f(x) = a(x - h)^2 + k, where (h, k) represents the coordinates of the vertex. This form is helpful in identifying the vertex and axis of symmetry.
3. Vertex:
To find the vertex, you need to convert the given equation into vertex form.
In your case, the given points P and Q lie on the parabola, so you can use them to form a system of equations:
P: (-2, 4) => f(-2) = 4 (1)
Q: (-1, 7) => f(-1) = 7 (2)
Plug the x-coordinate (-2) and (-1) into the quadratic function equation and solve for "a," "h," and "k".
Substituting (-2, 4) into equation (1):
4 = a(-2 - h)^2 + k (3)
Substituting (-1, 7) into equation (2):
7 = a(-1 - h)^2 + k (4)
You now have two equations with two unknowns, "a" and "h". Solving this system of equations will give you the values for "a" and "h".
4. Axis of Symmetry:
The axis of symmetry is a vertical line that divides the parabol