I'm working on graphing functions such as [math]x^2+3x[/math] with more than one x-value and really trying to understand how to "see" the graph from the equation/inequality before actually graphing it. I'm seeing some relationships, but wondering if there are any resources that can help. My book doesn't go into the intuition behind anything other than the basic function graphs (x^2, x^3, sqrt(x), x^2 - 3, (x+3)^2, etc.). But for equations like [math]x^2+3x[/math] it's a little more difficult.
I'm finding patterns such as for [math]x^2+3x[/math] the x-value of the vertex is at -1.5 (which is the negative midpoint of 0 and the coefficient 3 for x) and that the graph is below zero for all values -3 < x < 0. Likewise, for [math]x^2+4x[/math] the x-value of the vertex is -2 (which is the negative midpoint of 0 and the coefficient 4 for x) and the graph is below zero for all values -4 < x < 0.
I understand shifts like [math]x^2+3[/math] or [math](x+3)^2[/math], but I'm having difficulty understanding the relationship/pattern when another x-variable is added and also when there are higher order terms as well like [math]x^3+3x^2+2x+4[/math]. Any resources out there that discuss these well?
I'm finding patterns such as for [math]x^2+3x[/math] the x-value of the vertex is at -1.5 (which is the negative midpoint of 0 and the coefficient 3 for x) and that the graph is below zero for all values -3 < x < 0. Likewise, for [math]x^2+4x[/math] the x-value of the vertex is -2 (which is the negative midpoint of 0 and the coefficient 4 for x) and the graph is below zero for all values -4 < x < 0.
I understand shifts like [math]x^2+3[/math] or [math](x+3)^2[/math], but I'm having difficulty understanding the relationship/pattern when another x-variable is added and also when there are higher order terms as well like [math]x^3+3x^2+2x+4[/math]. Any resources out there that discuss these well?
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