How do we determine if a conic section has a rational point on it?
1 answer:
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<h2>Steps:</h2>
- Vertex Form: y = a(x - h)² + k with (h,k) as the vertex
So firstly, let's start with the vertex. Since this is in vertex form, we can find the vertex easily. Since 3 is in the h variable and -4 is in the k variable, <u>the vertex is (3,-4).</u>
Next, the axis of symmetry. Remember that the vertex's x-coordinate and the axis of symmetry are the same. In this case, since the vertex's x-coordinate is 3, this means that <u>the axis of symmetry is x = 3.</u>
Next, whether the vertex is a minimum or a maximum. To determine whether it's a minimum or a maximum, we look towards the a variable of the vertex form. If a is negative, then the parabola opens down and the vertex is a maximum. However, if a is positive, then the parabola opens up and the vertex is a minimum. In this case, a = 1 and since 1 is positive, <u>this makes the vertex a minimum.</u>
Next, to find the y-intercept plug 0 into the x-variable and solve:
<u>The y-intercept is (0,5).</u>
Next, to find the x-intercepts plug 0 into the y-variable to solve. Since it's a bit less straightforward than finding the y-intercept, I will walk through the steps:
Firstly, add 4 to both sides:
Next, square root both sides:
Next, add 3 to both sides:
Lastly, solve the left side twice: once with the plus sign, once with the minus sign:
<u>Your x-intercepts are (5,0) and (1,0).</u>
<h2>Answers:</h2>In short:
- Vertex: (3,-4)
- x-intercept(s): (5,0) and (1,0)
- y-intercept(s): (0,5)
- Axis of symmetry: x = 3
- Minimum or Maximum? Minimum.