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Find the vertex, axis of symmetry, y intercept of each parabola. State the number of x intercepts. Then sketch the graph of each function. f(x)=2(x-4)^2+4

Find the vertex, axis of symmetry, y intercept of each parabola. State the number-example-1

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Answer:

Vertex:
(x,y) = (4,4), Axis of symmetry:
x = 4, no x-Intercepts, y-Intercepts:
(x,y) = (0, 36). The graph is represented in the image attached below.

Explanation:

The equation of the parabola in vertex form and whose axis of symmetry is vertical is described by this formula:


y - k = C\cdot (x-h)^(2) (1)

Where:


x - Independent variable.


y - Dependent variable.


C - Vertex constant.


h, k - Coordinates of the vertex.

By direct comparison, we find the following information:


h = 4,
k = 4,
C = 2

Vertex

The vertex is a point of the parabola so that
(x,y) = (h, k).

If we know that
h = 4 and
k = 4, then the coordinates of the vertex are
(x,y) = (4,4).

Axis of symmetry

The axis of symmetry is a line of the form
x = h.

If we know that
h = 4, then the axis of symmetry is
x = 4.

To find the x and y intercepts, we need to transform the equation of the parabola into its standards, which is a second grade polynomial:


y - k = C\cdot (x-h)^(2)


y - k = C\cdot (x^(2)-2\cdot h\cdot x +h^(2))


y = C\cdot x^(2) -2\cdot C\cdot h \cdot x + (C\cdot h^(2)+k)

If we know that
h = 4,
k = 4 and
C = 2, then the equation of the parabola in standard form is:


y = 2\cdot x^(2)-16\cdot x + 36

x-Intercepts

The x-intercepts of the polynomial (if exist) can be found by the Quadratic Formula:


x_(1,2) = \frac{16\pm \sqrt{(-16)^(2)-4\cdot 2\cdot (36)}}{4}


x_(1,2) = 4 \pm (√(-32))/(4)


x_(1,2) = 4 \pm i \,(4√(2))/(4)


x_(1,2) = 4 \pm i\,√(2)

As both roots are conjugated complex numbers, there are no x-intercepts.

y-Intercepts

The y-intercept (if exists) can be found by evaluating the polynomial at
x = 0:


y = 2\cdot 0^(2) - 16\cdot 0 + 36


y = 36

The y-intercept is
(x,y) = (0, 36).

Lastly, we proceed to plot the function by using graphing tools.

Find the vertex, axis of symmetry, y intercept of each parabola. State the number-example-1
User Kaj Hejer
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