Question

At the horizontal intercept x = -3, coming from the (x+3) factor of the polynomial, the graph passes directly through the horizontal intercept. The factor (x+3) is linear (has a power of 1), so the behavior near the intercept is like that of a line - it passes directly through the intercept. We call this a single zero, since the zero corresponds to a single factor of the function. At the horizontal intercept x = 2, coming from the (x - 2)? factor of the polynomial, the graph touches the axis at the intercept and changes direction. The factor is quadratic (degree 2), so the behavior near the intercept is like that of a quadratic – it bounces off the horizontal axis at the intercept. Since (x - 2)= (x-2)(x-2), the factor is repeated twice, so we call this a double zero. We could also say the zero has multiplicity 2. At the horizontal intercept x =-1, coming from the (x+1)' factor of the polynomial, the graph passes through the axis at the intercept, but flattens out a bit first. This factor is cubic (degree 3), so the behavior near the intercept is like that of a cubic, with the same “S” type shape near the intercept that the toolkit xhas. We call this a triple zero. We could also say the zero has multiplicity 3. By utilizing these behaviors, we can sketch a reasonable graph of a factored polynomial function without needing technology. Graphical Behavior of Polynomials at Horizontal Intercepts If a polynomial contains a factor of the form (x-h)", the behavior near the horizontal intercept h is determined by the power on the factor. p=1 p=2 p=3 Single zero Multiplicity 1 Double zero Multiplicity 2 Triple zero Multiplicity 3 For higher even powers 4,6,8 etc.... the graph will still bounce off the horizontal axis but the graph will appear flatter with each increasing even power as it approaches and leaves the axis. For higher odd powers, 5,7,9 etc... the graph will still pass through the horizontal axis but the graph will appear flatter with each increasing odd power as it approaches and leaves the axis.

Answer 1

for example we graph the function f(x) = (x+3)(x-2)2 (2+1) ta 2 9 The function has graph with factors (*c+3), C-2) and (2+1) with degrees 1,2,3. → at each of the x- intercept the graph behaves differently based on the multiplicity 1. when wintercept OL=-3 of the solution of the graph x+3=o. since it act is a linear degree function (+3) The behaviour of the graph near the entercept is like of a kine

it is called as single 300, because zero correspondanc to single factor of the function. 1 single zero The x-intercept oc=2 has a second degree polynomial (-2)2. The graph touches the axis and changes the direction. It's behaviour near the axis is like of quadratic - it bounces of Zero with the horizontal amis at the multiplicity 2 entencept

for wintercept = -1 from (2-1)3 the graph passes through the axes at the intercept but flattens out a bit finst Thos factor is cubic, so the behaviour near the p=3 intercept is like that of a cubic with same s-shape near intercept as the funcios f(2)=13. the en zero with multiplicity Thus we can determine appearance the equation of a graph from its at horizontal intercept. pe en Fox zeroes with even values graph touch and bounce off x-axis, as the even value increaces the power the graph will appear flatter as it approaches and leaves zero For odd powers The graph crosses the x-ams, but as the odd power increaces, the graph appear flatter and as et appooaches and leaves the x-anis.