solving quadratic functions

Question

describe the graph in terms of transformations on the graph of y = x. Then sketch the graph. Label the vertex, axis of symmetry, and two other points

f(x) = -x^2 + 9

Answer 1

Given y = x^2

Transformation of graph y = -x^2+9

y = -x^2+9

Compare it to quadratic form ax^2+bx+c = 0

Axis of symmetry is x = b/2a = 0/2(1) = 0

Substitute the x value in y = -x^2+9

y = 0+9

y = 9

Vertex of the parabola is (x,y) = (0,9)

For other two points, let x = 1,-1 then y would be 8,8.

Coordinae pairs are (1,8),(-1,8).

Graph

From the graph

the parent function f(x) = x^2 vertex at origin.

y = -x^2+9 parabola graph opens downward and it's heighst point is vertex.

Axis of symmetry is 9.

Answer 2

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The function is f ( x ) = - x² + 9.

The heighest power of x in f ( x ) = - x² + 9 is 2, so the parent function is square function.

The equation of parent function is g ( x ) = x² .

The transformation function is f ( x ) = - x² + 9 and the parent function is g ( x ) = x² .

So, the functions f and g have the following relationship.

f ( x ) = - g ( x ) + 9

To transform the graph of g ( x ) = x² into the graph of  g(x) = f(x), first reflect the grafh of g ( x ) = x² in the x - axis.

Then, shift it to the upward nine units(vertical shift(upward) of nine units of the graph of g ( x ) = x²).

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The function is f(x) = y = - x² + 9.

The standard form of quadratic function is f(x) = ax² + bx + c.

Find the axis of symmetry :

Formula for the equation of the axis of symmetry :  x = - b/2a.

Substitute the values of b = 0 and a = - 1 in the formula, x = - b/2a.

x = - (0)/2(- 1)

x = 0.

The equation for the axis of symmetry is x = 0 (y - axis).

Find the vertex :

To find the vertex, use the value of equation for the axis of symmetry as the  x - coordinate of the vertex.

To find the y - coordinate, substitute the value of x = 0 in the original function, y = - x² + 9.

y = - (0)² + 9

y = 0 + 9

y = 9.

The vertex is (0, 9).

Determine whether the function has maximum or minimum value :

The value of a = - 1 < 0 (negative), so the graph of function opens downward and has a maximum value.

The maximum value (y - coordinate of the vertex) is 9.

Find other two points lies on the graph :

find the solution point of the two functions f (x) = y =  - x² + 9 and g ( x ) = y = x² .

Substitute y = x²  in y =  - x² + 9.

y =  - y + 9

2y = 9

⇒ y = 9/2 = 4.5.

To solve for x, substitute y = 4.5 in either of two functions.

4.5 = x²

x = ± √4.5

⇒ x = ± 2.21.

Therefore, the solution points are (2.21, 4.5) and (- 2.21, 4.5).

These are other two points should lies on the graphs of both f (x) = - x² + 9 and g ( x ) = x² .

Graph :