Question

Find the x intercept of the parabola with vertex 3,-2 and y intercept 0,7 round to the nearest hundredth

Answer 1

The x-intercepts of the given parabola are approximately x = 3.41 and x = 2.59.

Step-by-step explanation:

The x-intercept of a parabola can be found by setting the y-coordinate to zero and solving for the x-coordinate.

Given that the vertex of the parabola is at (3, -2) and the y-intercept is at (0, 7), we can use the vertex form of a parabolic equation: y = a(x - h)^2 + k

Substituting the vertex coordinates into the equation, we have: y = a(x - 3)^2 - 2

Since the y-intercept is at (0, 7), we can substitute these coordinates into the equation to solve for the value of a:

7 = a(0 - 3)^2 - 2

Simplifying the equation gives: 7 = 9a - 2

Adding 2 to both sides gives: 9a = 9

Dividing both sides by 9 gives: a = 1

Now that we know the value of a, we can substitute it back into the equation to find the x-intercepts:

0 = 1(x - 3)^2 - 2

Simplifying the equation gives: (x - 3)^2 = 2

Taking the square root of both sides gives: x - 3 = ±√2

Adding 3 to both sides gives: x = 3 ± √2

Rounding to the nearest hundredth, the x-intercepts are approximately x ≈ 3.41 and x ≈ 2.59.

Answer 2

an x-intercept is namely a "solution" or "zero" or "root" often called, and when that happens, y = 0, just like with any other x-intercept.


`\bf ~~~~~~\textit{parabola vertex form} \\\\ \begin{array}{llll} \boxed{y=a(x- h)^2+ k}\\\\ x=a(y- k)^2+ h \end{array} \qquad\qquad vertex~~(\stackrel{}{ h},\stackrel{}{ k})\\\\ -------------------------------\\\\ \begin{cases} h=3\\ k=-2 \end{cases}\implies y=a(x-3)^2-2 \\\\\\ \textit{we also know that } \begin{cases} x=0\\ y=7 \end{cases}\implies 7=a(0-3)^2-2 \\\\\\ 9=9a\implies \cfrac{9}{9}=a\implies 1=a\qquad therefore\qquad \boxed{y=(x-3)^2-2}`

so what is its x-intercept anyway?


`\bf \stackrel{y}{0}=(x-3)^2-2\implies 2=(x-3)^2\implies \pm√(2)=x-3 \\\\\\ \pm√(2)+3=x\qquad therefore\qquad (\pm√(2)+3~~~~,~~~~0)`