SOLUTION: What is the domain of f, range of f, vertex, axis of symmetry, x-intercept (s), y-intercept, determine whether it has a minimum or a maximum value and state that value for f(x)=x^2

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Question 190301: What is the domain of f, range of f, vertex, axis of symmetry, x-intercept (s), y-intercept, determine whether it has a minimum or a maximum value and state that value for f(x)=x^2+x-2
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Since there are no real number values for which this function is undefined the domain is all real numbers or .

Since this is a quadratic polynomial, the graph is a parabola. Since the lead coefficient is positive, the parabola opens upward. Let's table discussion of the Range until we have identified the vertex.

The x-coordinate of the vertex of a parabola described by a function in the form:




is given by:

so for your function:

The y-coordinate of the vertex of such a parabola is given by



So the vertex is the point

Since this is a parabola opening upward, the y-coordinate of the vertex represents the minimum value of the function. And there is no maximum value, so we can now define the range as [,). This also answers the question about whether there is a minimum or maximum -- and what the value of the minimum is.

The axis of symmetry is the vertical line defined by the equation where a is the value of the x-coordinate of the vertex, so:



The y-intercept is the value of the function when x equals zero, so:



The x-intercepts are the values of x that make the value of the function equal zero, so:



The trinomial factors, so:











John