Find the roots, y-intercept, and vertex of the quadratic function f(x) = x2 − 2x − 3.

see explanation

Step-by-step explanation:

To find the roots let f(x) = 0, that is

x² - 2x - 3 = 0

(x - 3)(x + 1) = 0 ← in factored form

Equate each factor to zero and solve for x

x - 3 = 0 ⇒ x = 3

x + 1 = 0 ⇒ x = - 1

Roots are x = - 1, x = 3

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To find the y- intercept let x = 0, that is

f(0) = 0 - 0 - 3 = - 3 ← y- intercept

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The vertex lies on the axis of symmetry which is situated at the midpoint of the roots.

= = 1

substitute x = 1 into f(x) for y- coordinate of vertex

= 1² - 2(1) - 3 = 1 - 2 - 3 = - 4

vertex = (1, - 4)

35x 3= 105

step-by-step explanation:

35+35= 70   70 + 35= 105

let f (x ) is a function and c be a positive real number,   the graph of the function g (x ) = f (cx ) is represented as follows.

when, c > 1,   the function compresses it in the x - direction.

when, 0 < c < 1, the function stretches it.

the function is g ( x ) = (9x )2.

the heighest power of x in g ( x ) = (9x )2 is 2, so the parent function is square function.

the equation of parent function is f ( x ) = x 2 .

the function is g ( x ) = (9x )2 and the parent function is f ( x ) = x 2 .

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

g ( x ) = f ( 9x ).

compare the function g ( x ) = f ( 9x ) with g (x ) = f (cx ).

c = 9 ( > 1 ).

so, the function g (x ) = (9x )2 compresses it in the x - direction.

step-by-step explanation: