Find the Properties f(x)=x^2-2x-3

Problem

ƒ(x)=x2−2*x−3

Solution

1. Identify the coefficients of the quadratic function in the form ƒ(x)=a*x2+b*x+c

a=1

b=−2

c=−3

2. Determine the direction of opening by looking at the sign of a Since a>0 the parabola opens upward.

3. Find the vertex (h,k) using the formula h=(−b)/(2*a)

h=(−(−2))/(2*(1))=1

k=ƒ(1)=(1)2−2*(1)−3=−4

Vertex=(1,−4)

4. Identify the axis of symmetry, which is the vertical line passing through the vertex.

x=1

5. Find the y-intercept by evaluating ƒ(0)

ƒ(0)=0−2*(0)−3=−3

y-intercept=(0,−3)

6. Find the x-intercepts by setting ƒ(x)=0 and factoring the quadratic.

x2−2*x−3=0

(x−3)*(x+1)=0

x=3,x=−1

x-intercepts=(3,0),(−1,0)

7. Determine the domain and range. The domain of any quadratic is all real numbers. Since the parabola opens upward from the vertex (1,−4) the range is all values greater than or equal to −4

Domain=(−∞,∞)

Range=[−4,∞)

Final Answer

Properties: Vertex *(1,−4), Axis of Symmetry *x=1, y-intercept *(0,−3), x-intercepts *(3,0)* and *(−1,0)

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