Question

A quadratic function f is given. flx) = -x2. 4x 3 (a) Express f in transformation form. f(x) (b) Find the vertex and x and y-intercepts of f. (If an answer does not exist, enter DNE.) (x, y) vertex (x, y) (smaller x-value) x-intercepts (x, y) (larger x-value) (x, y) yintercept (c) Sketch a graph of f. Tools Actions Clear Al 1. Select an object from the Tools menu to the left. Delato 2. Enter coordinates in Object Properties below, or use the mouse to place and move objects. To enter a fractional or decimal coordinate, use Object Properties. No Solution view our tutorial videos Object Properties Select a Tool or Object to Begin Help WebAssign graphing tool (d) Find the domain and range of f. (Enter your ansswers using interval notation.) domain range

Answer 1

a) f(x) = -x² + 4x + 3

= - (x² - 4x) + 3

= - (x² - 2x(2) + (2)² - (2)²) + 3

= - (x² - 2x(2) + (2)²) + (2)² + 3

= - (x - 2)² + 7

Therefore,

f(x) = - (x - 2)² + 7

b) Vertex (x, y) = (2, 7)

To find the x-intercepts, let us equate f(x) = 0

=> - (x - 2)² + 7 = 0

=> (x - 2)² - 7 = 0

=> (x - 2 + sqrt(7)) (x - 2 - sqrt(7)) = 0

=> x = 2 - sqrt(7), x = 2 + sqrt(7)

Therefore,

x-intercepts (x, y) = (2 - sqrt(7), 0) (smaller x- value)

(x,y) = (2 + sqrt(7), 0) (larger x-value)

To find the y-intercept, let us substitute x = 0 in f(x)

f(0) = - (0 - 2)² + 7 = 3

Therefore,

y-intercept (x,y) = (0, 3)

c) The graph of f is as below:

2 0 2 -6 CO CC st

d) Domain is the set of real values of x for which the function is real and defined.   

In this case, since f(x) is real and defined for all the real values of x, therefore

Domain is (- $\infty$, $\infty$)

Range is the set of real values the function can take in the defined domain.

From the graph, we observe that the value of the function f(x) is always less than or equal to 7.

Range is (- $\infty$, 7]

Answer 2

A quadratic function f is given. flx) = x^2+4x-3 (a) Express f in transformation form