How do you find the domain and range of #y=x^2 +2x -5#?

Question

Kennedy Adams · Accepted Answer

Domain: #(-oo, +oo)#
Range: #[-6, +oo)# #y=x^2+2x-5# #y# is defined #forall x in RR# Hence the domain of #y# is #(-oo, +oo)# #y# is a quadratic function of the form #ax^2+bx+c# The graph of #y# is a parabola with vertex where #x=(-b)/(2a)# Since the coefficient of #x^2>0# the vertex will be the absolute minimum of #y# At the vertex #x= (-2)/(2xx1) = -1# #:. y_min = y(-1) = 1-2-5 = -6# Since #y# has no upper bound the range of #y# is [-6, +oo) As can be seen on the graph of #y# below. x^2 + 2x-5 = [-16.02, 16.02, -8.01, 8.01]}

Nathan Axel · Answer

undefined To find the domain and range of the function $ y = x^2 + 2x - 5 $, you need to consider the possible values of $ x $ that make the function defined, and the corresponding values of $ y $ that the function can take.

**Domain:**
Since $ y = x^2 + 2x - 5 $ is a polynomial function, it is defined for all real numbers. Therefore, the domain of the function is all real numbers.

**Range:**
To find the range, you can analyze the behavior of the quadratic function. The function $ y = x^2 + 2x - 5 $ is a quadratic function, and its graph is a parabola opening upwards because the coefficient of $ x^2 $ is positive.

The minimum or maximum value of the quadratic function occurs at the vertex of the parabola. You can find the x-coordinate of the vertex using the formula $ x = \frac{-b}{2a} $, where $ a $ and $ b $ are the coefficients of $ x^2 $ and $ x $ respectively in the quadratic equation $ y = ax^2 + bx + c $.

In this case, $ a = 1 $ and $ b = 2 $, so $ x = \frac{-2}{2} = -1 $.

Now, substitute $ x = -1 $ into the function to find the corresponding value of $ y $:
$ y = (-1)^2 + 2(-1) - 5 = 1 - 2 - 5 = -6 $

So, the vertex of the parabola is at the point $ (-1, -6) $. Since the parabola opens upwards, the minimum value of $ y $ occurs at the vertex, and the range of the function is all real numbers greater than or equal to the y-coordinate of the vertex.

Therefore, the range of the function $ y = x^2 + 2x - 5 $ is $ y \geq -6 $.