1. **Problem 5:** Identify which graph represents the polynomial function $f(x) = 2x^3 - 5x^2 + 4x + 1$. - The function is a cubic polynomial with leading coefficient positive ($2$), so as $x \to \infty$, $f(x) \to \infty$ and as $x \to -\infty$, $f(x) \to -\infty$. - It can have up to 3 real roots and up to 2 turning points. - Graph A shows a cubic curve with one local maximum and one local minimum, crossing the x-axis three times, matching the behavior of $f(x)$. - Graph B is a parabola (quadratic), Graph C has vertical asymptotes (not polynomial), Graph D is linear. **Answer:** a. Graph A 2. **Problem 6:** How do the coefficients in $f(x) = 2x^3 - 5x^2 + 4x + 1$ influence the graph? - The coefficients affect the steepness (how quickly the graph rises or falls). - They also influence the number and location of x-intercepts and turning points. - The leading coefficient ($2$) determines the end behavior. **Correct choice:** b. They determine the steepness of the curve 3. **Problem 7:** Relationship between degree and turning points. - A polynomial of degree $n$ can have up to $n-1$ turning points. - Higher degree polynomials generally have more turning points. **Correct choice:** b. Higher degree polynomials have more turning points 4. **Problem 8:** Effect of leading coefficient on end behavior. - The sign and magnitude of the leading coefficient determine whether the ends of the graph go up or down and how steeply. **Correct choice:** c. It determines the end behavior of the graph 5. **Problem 9:** Find $H(2)$ for $H(t) = 2t^3 - 5t^2 + 3t + 1$. Calculate: $$H(2) = 2(2)^3 - 5(2)^2 + 3(2) + 1 = 2(8) - 5(4) + 6 + 1 = 16 - 20 + 6 + 1 = 3$$ **Answer:** b. 3 6. **Problem 10:** If growth rate doubles and there is an additional constant height increase of 2 cm every week, how does $H(t)$ change? - Doubling growth rate means multiply the polynomial terms except the constant by 2: $$2 \times (2t^3 - 5t^2 + 3t) = 4t^3 - 10t^2 + 6t$$ - Additional constant height increase of 2 cm every week means adding $2t$ to the function. - So new function: $$H_{new}(t) = 4t^3 - 10t^2 + 6t + 1 + 2t = 4t^3 - 10t^2 + 8t + 1$$ None of the options exactly match this, but closest is option d with constant term 2 instead of 1 and no extra $2t$ term. Since the problem states "additional constant height increase of 2 cm every week," it means adding $2t$, so the function should be: $$H_{new}(t) = 4t^3 - 10t^2 + 8t + 1$$ No option matches exactly, but option d is closest in doubling terms and adding constant 2 (though constant should be 1). **Answer:** d. $H(t) = 4t^3 - 10t^2 + 6t + 2$ (closest match) --- "slug": "polynomial graphs", "subject": "algebra", "desmos": {"latex": "y=2x^3-5x^2+4x+1","features": {"intercepts": true,"extrema": true}}, "q_count": 6