a) vertex<\/p>\n
b) domain and range<\/p>\n
c) axis of symmetry<\/p>\n
d) max or min value (and tell whether it is a max or a min)<\/p>\n
e) y intercept<\/p>\n
f) x intercept(s), if any<\/p>\n
Solution: Let’s assume we are going to use factored form. (An alternative is vertex form; I’ll tackle that method in another post.)<\/p>\n
Our first step is to factor x^2 – 6x – 16 (you might want to review my article here<\/a> on factoring easy trinomials).<\/p>\n We seek the numbers that multiply to make -16, but add to make -6. Realizing they are -8 and 2, we write<\/p>\n y=x^2 – 6x -16 \u21d2 y=(x-8)(x+2)<\/p>\n Looking at the factored form, we notice that when x=8 or x=-2, y must be 0. Therefore, the points (8,0) and (-2,0) must be on the graph.<\/p>\n In the original function y=x^2 – 6x -16, note that the coefficient of x^2 is positive. Therefore, this graph will have a minimum value; from the vertex it will rise. Knowing this, and that it includes (8,0) and (-2,0), we can make a quick sketch of the graph:<\/p>\n Midway between the two x-intercepts, aka zeros, (-2,0) and (8,0), lies the vertex. The middle between -2 and 8 is found thus:<\/p>\n middle=(-2+8)\/2 = 6\/2 = 3<\/p>\n Therefore, the vertex occurs at x=3. To find the y coordinate, we plug 3 in for x in the original equation:<\/p>\n y=3^2 – 6(3) -16 = 9-18-16=-25<\/p>\n Therefore, the vertex is at (3,-25).<\/p>\n The domain of a quadratic function is all real numbers unless it’s a word problem with real-life limitations. In this case, the domain is all real numbers.<\/p>\n The range is y\u2265-25, which you can see from the graph when you place (3,-25) in for the vertex.<\/p>\n The axis of symmetry is x=3; it’s the vertical line that cuts the graph down the middle. For a quadratic function with vertex (p,q), the axis of symmetry is x=p.<\/p>\n The graph has a min; its min value is -25. The max or min value of a quadratic function with vertex (p,q) is q.<\/p>\n To find the y intercept, set x to 0. Doing that in the original function, we see y=-16.<\/p>\n We’ve already found the x intercepts: they’re at x=-2 and x=8.<\/p>\n While this article is a good primer, there’s still more to mention. In fact, I’ll be saying much more about quadratic functions in future posts.<\/p>\n HTH:)<\/p>\n![](\"\/blog\/blogfiles\/quadgrph.png\")
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