Grade 11 · Polynomial roots · 64 of 600
Polynomial roots · First root a=0.5; Second root b=0.5
Evaluate at x=0.8. Find the function value, tangent slope, and vertical line of symmetry. Givens: First root a=0.5; Second root b=0.5.
Try the question, use a hint when you need one, and compare your reasoning with the worked solution. The animation starts with this chapter’s values; changing its controls explores a new case.
01 · Make a prediction
Your practice question
A monic polynomial is f(x)=(x−(0.5))(x−(0.5)). Evaluate at x=0.8. Find the function value, tangent slope, and vertical line of symmetry.
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A starting point
Evaluate both factors before multiplying. The derivative of x²−(a+b)x+ab is 2x−a−b.
Work through the reasoning
Step 1
Identify the model and target
A monic polynomial is f(x)=(x−(0.5))(x−(0.5)). Evaluate at x=0.8. The governing relation is f(x)=(x−a)(x−b); f′(x)=2x−a−b. Evaluate both factors before multiplying. The derivative of x²−(a+b)x+ab is 2x−a−b.
Step 2
Substitute and calculate
Substitute x=0.8: (0.8−(0.5))(0.8−(0.5))=0.09. Differentiating gives 2(0.8)−(0.5)−(0.5)=0.6.
Step 3
Check the mathematical meaning
The symmetry line is x=(0.5+(0.5))/2=0.5. The derivative there is zero. At x=0.8 the signs of the two factors must agree with the sign of 0.09. Input x: 0.8; Product value: 0.09; Slope: 0.6; Symmetry line x: 0.5. Decimal values are rounded, so use unrounded intermediate values.
The answer
Substitute x=0.8: (0.8−(0.5))(0.8−(0.5))=0.09. Differentiating gives 2(0.8)−(0.5)−(0.5)=0.6. The symmetry line is x=(0.5+(0.5))/2=0.5. The derivative there is zero. At x=0.8 the signs of the two factors must agree with the sign of 0.09. Animation check: Input x: 0.8; Product value: 0.09; Slope: 0.6; Symmetry line x: 0.5. Decimal displays are rounded; retain the original parameters and exact π until the final step.
Compare the method as well as the result. A different valid method may reach the same answer. Keep exact values until the last step when the question asks for rounding.
02 · Explore the model
See the mathematical relationship move.
Use Play, Pause, and the timeline to inspect the construction. Reset restores the question’s original settings. The displayed assumptions describe where this model applies.
Watch the relationship
PausedHD animation studio
From experiment to screen.
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The mathematical idea
Factored form locates the zeros before any expansion. Two different roots create two crossings, whereas coincident roots produce a repeated factor and a touching point. The moving input exposes how the signs of the two factors determine the graph's sign between and outside the roots. A monic polynomial is f(x)=(x−(0.5))(x−(0.5)). Evaluate at x=0.8. The requested state occurs at 60% playback; use the exact target stated in the question for your calculation.
f(x)=(x−a)(x−b); f′(x)=2x−a−b
03 · Reflect and transfer
Explain what changes and why.
Move the input across each root. How do distinct roots and a repeated root produce different sign changes?
This is a distinct guided scenario using a reusable mathematical model. Similar-looking diagrams can represent different given values and conclusions; they are not different mathematical theories.