sketch each parabola using the given information. vertex (2,3) point (6,9)
The canonical form of the parabola is
y=a(x-h)²+k
where (h,k) is the vertex, and a is a constant.
For a vertex of (2,3), we have
y=a(x-2)²+3
where a is a constant to be found from the other point, (6,9)
=>
9=a(6-2)²+3
=>
a=3/8
The complete equation of the parabola is therefore
y=(3/8)(x-2)²+3
Check:
y(6)=9 OK
y(2)=3 OK
y'(x)=(3(x-2)^2)/8+3
y'(2)=0 OK
For sketching, it would help to know the zeroes (if any), so put
y(x)=(3/8)(x-2)²+3=0
=>
(3/8)(x-2)²=-3
=>
there are no real zeroes.
Did you hear about the party that the parabola was throwing? It was all about the vertex, man! So, let's get this sketch party started!
The vertex of the parabola is given as (2,3), and we also have another point on the parabola (6,9).
Step 1: Plot the vertex of the parabola, which is (2,3). Let's call this point A.
Step 2: Plot the other given point, (6,9). We'll call this point B.
Step 3: Draw a line segment connecting points A and B. This line segment will be the axis of symmetry of the parabola.
Step 4: From the vertex point A, measure the same distance horizontally and vertically to points on either side of the axis of symmetry. Let's say we measure a distance of 2 units in both directions. So, two units to the right of point A, we'll label as point C. Similarly, two units to the left of point A, we'll label as point D.
Step 5: We'll draw a line segment from points C and D to point B. And voila! We have our parabola.
Now, keep in mind that this is just a sketch. It's not perfectly to scale, but it gives you a visual idea of what the parabola might look like based on the given information. Enjoy the party!
To sketch a parabola using the given information, we need to find the equation of the parabola.
Step 1: Finding the equation in vertex form
The vertex form of a parabolic equation is given by:
y = a(x - h)^2 + k
where (h, k) represents the coordinates of the vertex.
In this case, the vertex is (2, 3). So, we have:
y = a(x - 2)^2 + 3
Step 2: Finding the value of 'a'
To find the value of 'a', we can use the given information of another point on the parabola, which is (6, 9). Substituting these coordinates into the equation, we get:
9 = a(6 - 2)^2 + 3
Simplifying the equation:
9 = a(4)^2 + 3
9 = 16a + 3
16a = 9 - 3
16a = 6
a = 6/16
a = 3/8
So, the equation becomes:
y = (3/8)(x - 2)^2 + 3
Step 3: Sketching the parabola
Using the equation y = (3/8)(x - 2)^2 + 3, we can plot the points and sketch the parabola:
- Plot the vertex: (2, 3)
- Plot the given point: (6, 9)
- Find one more point on the parabola by substituting a value of 'x' and solving for 'y'. For example, when x = 0:
y = (3/8)(0 - 2)^2 + 3
y = (3/8)(-2)^2 + 3
y = (3/8)(4) + 3
y = 3/2 + 3
y = 6/2 + 3
y = 3 + 3
y = 6
The point is (0, 6).
- Connect the points to form a smooth curve.
The sketch should look like a U-shaped curve opening upwards, passing through the points (2, 3), (6, 9), and (0, 6).
To sketch the parabola using the given information, we can follow these steps:
1. Determine the vertex form equation of a parabola:
The vertex form equation of a parabola is given by: y = a(x - h)^2 + k, where (h, k) represents the vertex of the parabola.
2. Use the given vertex and the point to determine the value of 'a':
Since the vertex of the parabola is (2,3), we can substitute these values into the vertex form equation to get: 3 = a(2 - 2)^2 + 3. Simplifying this equation, we find that a = 1.
3. Plug in the value of 'a' into the vertex form equation:
Now that we have the value of 'a', we can write the equation of the parabola as: y = (x - 2)^2 + 3.
4. Plot the vertex and the given point:
The vertex is located at (2, 3). Plot this point on a coordinate grid. The given point is (6, 9). Plot this point as well.
5. Sketch the parabola:
Using the equation y = (x - 2)^2 + 3, we can plot additional points on the graph by substituting different values of x into the equation. For example, when x = 1, y = (1 - 2)^2 + 3 = 2. Similarly, when x = 3, y = (3 - 2)^2 + 3 = 4. Plot these points and connect them on the graph. Continue this process to sketch the parabola.
6. Label the graph:
Label the x-axis, y-axis, and the vertex on the graph to complete the sketch.
By following these steps, you will be able to sketch the parabola with the given information.