find the x intercepts of a parabola whenthe vertex is at 2,13 and y intercept at 0,5
the generic equations of parabola is:
y=ax^2+bx+c
by information of the vertex 2;13 i have two facts:
i) 13=a*2^2+b*2+c ....replacing(2;13)
ii)the director of the vertex as equation:
x=-b/2a
by information "intercept at 0,5 ":
iii)5=a*0^2+b*0+c ....replacing(0;5)
tre equation tre unknowns a,b,c
you solve the system
i forgot to specify in ii) x=-b/(2a) => 2=-b/(2a)
To find the x-intercepts of a parabola, we need to determine the values of x when y is equal to zero. In this case, let's start by identifying the equation of the parabola.
Since we know the vertex is at (2, 13), we can use the vertex form of a parabola equation, which is given as follows:
y = a(x - h)^2 + k,
where (h, k) represents the vertex.
By substituting the vertex coordinates into the equation, we have:
y = a(x - 2)^2 + 13.
To find the value of 'a', we can use the y-intercept, which is (0, 5). Substituting these coordinates into the equation, we get:
5 = a(0 - 2)^2 + 13,
5 = 4a + 13,
4a = -8,
a = -2.
Now that we know the value of 'a' is -2, we can rewrite the equation of the parabola as:
y = -2(x - 2)^2 + 13.
Next, we can set y = 0 and solve the equation to find the x-intercepts:
0 = -2(x - 2)^2 + 13.
To simplify, let's subtract 13 from both sides:
-13 = -2(x - 2)^2.
Dividing both sides by -2:
6.5 = (x - 2)^2.
Taking the square root of both sides:
√6.5 = x - 2.
To isolate 'x', we add 2 to both sides:
√6.5 + 2 = x.
Now, we can obtain the approximate x-intercepts by calculating the value:
x ≈ √6.5 + 2 ≈ 4.55 or x ≈ -√6.5 + 2 ≈ -0.55.
Therefore, the x-intercepts of the parabola with the given vertex and y-intercept are approximately 4.55 and -0.55.