Part 1Write a polynomial function whose graph is shown (use the smallest degree possible.) -5-4-3-2-112345-8-6-4-22468xyleft parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458left parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458left parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458left parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458 Edit coordinates (0,0) A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 0.5 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. From left to right, a curve falls in quadrant 2 , crossing the x dash axis at x equals negative 4 to a minimum in quadrant 3, rises to a maximum at the point left parenthesis negative 2, 0 right parenthesis, falls and crosses the y dash axis at y equals negative 4 to a minimum in quadrant 4, rises at an increasing rate, crossing the x dash axis at x equals 2, and rises steeply in quadrant 1. The curve passes through the plotted point (1, negative StartFraction 45 Over 8 EndFraction). Part 1f(x)equals=[input]one eighth left parenthesis18( (Type your answer in factored form. Use integers or fractions for any numbers in the expression.)

Question

Part 1Write a polynomial function whose graph is shown​ (use the smallest degree​ possible.) -5-4-3-2-112345-8-6-4-22468xyleft parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458left parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458left parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458left parenthesis 1,negative StartFraction 45 Over 8 EndFraction right parenthesis1,−458 Edit coordinates (0,0) A coordinate system has a horizontal x-axis labeled from negative 5 to 5 in increments of 0.5 and a vertical y-axis labeled from negative 8 to 8 in increments of 1. From left to right, a curve falls in quadrant 2 , crossing the x dash axis at x  equals negative 4  to a minimum in quadrant 3, rises to a maximum at the point left parenthesis negative 2, 0 right parenthesis, falls and crosses the y dash axis at y  equals negative 4  to a minimum in quadrant 4, rises at an increasing rate, crossing the x dash axis at x  equals 2, and rises steeply in quadrant 1. The curve passes through the plotted point (1, negative StartFraction 45 Over 8 EndFraction). Part 1​f(x)equals=[input]one eighth left parenthesis18( ​(Type your answer in factored form. Use integers or fractions for any numbers in the​ expression.)

BlackTom AI · Accepted Answer

The task asks for a polynomial function whose graph matches a described curve and passes through the plotted point (1, -45/8), with the smallest possible degree.

Option under analysis: f(x) = (3/8)(x + 4)(x + 2)(x - 2).

First, identify the zeros from the factors: x = -4, x = -2, and x = 2. This aligns with the description that the curve crosses the x-axis at x = -4 and x = 2, and also crosses again at x = -2 (consistent with a vertex at (-2, 0) in the stated diagram). The multiplicities indicated here are all 1, so the graph will cross the x-axis at each intercept rather than merely touch it.

Next, evaluate the y-value at x = 1 to verify the given point (1, -45/8):
f(1) = (3/8)(1 + 4)(1 + 2)(1 - 2) = (3/8)(5)(3)(-1) = (3/8)(-15) = -45/8.
This exactly matches the plotted point, confirming that the scale factor 3/8 is chosen to fit the curve through that point.

Consider the end behavior and rough shape: for large negative x, each factor is negative except the (x+4) term becomes negative, (x+2) negative, (x-2) negative, so the product of three negatives is negative; multiplying by a positive 3/8 yields negative infinity on the left, which is consistent with the curve entering quadrant II and heading downward as x decreases. For large positive x, all three factors are positive, giving a positive value that rises steeply, matching the described rise in quadrant I. Thus the overall end behavior and turning structure are compatible with the description of the graph.

Why this option is correct in the given context: it produces zeros at the required x-intercepts, passes through the specified point (1, -45/8), and has the smallest degree (a cubic) that accommodates three distinct real zeros as described. The chosen leading coefficient 3/8 scales the curve to meet the exact y-value at x = 1, ensuring the graph passes through the plotted point.

Why alternative ideas (in general) might be misleading: if one tried a different leading coefficient without recomputing to hit (1, -45/8), the point (1, -45/8) would not be satisfied, so the graph would misfit that key checkpoint. If any intercepts were assumed incorrectly (e.g., missing a root or assigning a higher multiplicity), the shape near the intercepts would not cross the axis as described, leading to a mismatch with the quadrants and vertex positions shown. In short, the combination of roots and the scaling factor is precisely what makes this polynomial fit both the intercepts and the given point, while preserving the smallest possible degree.

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