Question

What are the coordinates for the vertex, and what are the x-intercepts?

Answer 1

To find the x-intercept(s), we can use the quadratic formula:

`x=\frac{-b\pm\sqrt[]{b^2-4ac}}{2a}`

This formula is applied when we have a polynomial of the form:

`ax^2+bx+c=0`

We have that our case is:

`(1)/(4)x^2+(3)/(2)x-(1)/(8)=0`

To find the x-intercepts of the parabola. However, we can find the least common multiple of the denominators to have an easier equation to solve. Then, we have:

Then, we can multiply each side of the equation by 8 as follows:

`8((1)/(4)x^2+(3)/(2)x-(1)/(8)=0)`

`(8)/(4)x^2+(8\cdot3)/(2)x-(8\cdot1)/(8)=8\cdot0\Rightarrow2x^2+12x-1=0`

Now, we have that:

a = 2

b = 12

c = -1

Then, we have:

`x=\frac{-12\pm\sqrt[]{12^2^{}-4(2)(-1)}}{2\cdot2}`

`x=\frac{-12\pm\sqrt[]{144+8}}{4}\Rightarrow x=\frac{-12\pm\sqrt[]{152}}{4}`

Then

`\sqrt[]{152}=\sqrt[]{2^2\cdot2\cdot19}=2\cdot\sqrt[]{38}`

Then, the two x-intercepts are:

`x=\frac{-12\pm2\cdot\sqrt[]{38}}{4}`

`x=\frac{-12+2\cdot\sqrt[]{38}}{4}\Rightarrow x=-(12)/(4)+\frac{2\cdot\sqrt[]{38}}{4}=-3+\frac{\sqrt[]{38}}{2}`

And

`x=\frac{-12-2\cdot\sqrt[]{38}}{4}\Rightarrow x=-(12)/(4)-\frac{2\cdot\sqrt[]{38}}{4}=-3-\frac{\sqrt[]{38}}{2}`

Therefore:

The smallest x-intercept is:

`-3-\frac{\sqrt[]{38}}{2}`

The largest x-intercept is:

`-3+\frac{\sqrt[]{38}}{2}`

Finding the value of the vertex

To find the vertex of the parabola, we can find its x-value, and y-value using the following formulas:

`x_v=-(b)/(2a),y_v=c-(b^2)/(4a)_{}`

In the original equation, we have:

a = 1/4

b = 3/2

c = -1/8

Then, we have:

`x_v=-\frac{(3)/(2)}{2((1)/(4))_{}}=-((3)/(2))/((2)/(4))=-(3)/(2)(4)/(2)=-(3\cdot4)/(4)\Rightarrow x_v=-3`

And

`y_v=-(1)/(8)-(((3)/(2))^2)/(4((1)/(4)))=-(1)/(8)-((9)/(4))/(1)=-(1)/(8)-(9)/(4)=-((1)/(8)+(9)/(4))=-((4+72)/(32))`

Finally:

`y_v=-(76)/(32)\Rightarrow y_v=-(19)/(8)`

Therefore, the vertex of the parabola is:

`(-3,-(19)/(8))`

In summary, we have:

The coordinates for the vertex are:

`(-3,-(19)/(8))`

The smallest x-intercept (only the