1. Problem: Analyze the polynomial $f(x) = (x + 1)(x - 1)(x - 3)$ of degree 3. - Sum of coefficients: Expand and calculate the values. - Even or Odd Degree: 3 (odd). - End behavior: As $x \to \infty, f(x) \to \infty$; as $x \to -\infty, f(x) \to -\infty$ because leading term is $x^3$. - Number of Turning Points: Maximum $2$ (degree - 1). - X-intercepts: Solve $f(x) = 0$ at $x = -1, 1, 3$. - Y-intercept: Evaluate $f(0) = (1)(-1)(-3) = 3$. - Graph: S-shaped cubic curve crossing x-axis at roots. 2. For $f(x) = (x + 1)^2 (x - 3)^2$, degree 4: - Sum coefficients found by expanding or plugging $x=1$. - Degree even: $4$. - Ends both up as leading $x^4$ positive. - Turning points max $3$. - X-intercepts: $-1$ (multiplicity 2), $3$ (multiplicity 2). - Y-intercept: $f(0) = (1)^2(-3)^2=9$. - Graph: W-shaped, touches x-axis at roots. 3. $f(x) = x + 6x^3 + 9x^2 - 4x - 12$ simplify to $6x^3+9x^2 -3x -12$. - Degree 3, odd. - Ends: left down, right up. - Turning points max 2. - X-intercepts: Solve $6x^3 + 9x^2 - 3x - 12=0$. - Y-intercept: $f(0)=-12$. 4. $f(x) = (x+1)^2 (x-3)$ degree 3 odd. - Sum coefficients expand or plug $x=1$. - Ends as cubic odd degree. - Turning points max 2. - Roots at $-1$ (multiplicity 2), $3$. - Y-intercept $f(0) = 1^2*(-3) = -3$. 5. $f(x) = (x-2)^2 - 5$ is quadratic transformed. - Degree 2 even. - Ends up (if positive leading coefficient). - Turning points 1. - Root solve $(x-2)^2=5$, $x=2 \pm \sqrt{5}$. - Y-intercept $f(0) = 4 - 5 = -1$. 6. $f(x) = -x^2 - 3x - 4$, quadratic degree 2. - Sum coefficients $-1 - 3 - 4 = -8$. - Ends down (leading coefficient negative). - Turning points 1. - X-intercepts from quadratic formula. - Y-intercept $f(0)=-4$. 7. $f(x) = 5x^2$, quadratic, degree 2. - Sum coefficients 5. - Ends up. - Turning points 1. - X-intercept at 0. - Y-intercept 0. 8. $f(x) = x^3 + 2x^2 - 3x$, degree 3. - Sum coefficients $1 + 2 - 3 = 0$. - Ends cubic odd. - Turning points 2. - Roots at $x(x^2 + 2x -3)=0$, roots $0$, $-3$, and $1$. - Y-intercept 0. 9. $f(x) = x^4 - 2x^2 + 1$, degree 4. - Sum coef $1 - 2 + 1 =0$. - Ends up. - Turning points max 3. - X-intercepts solve quartic. - Y-intercept 1. 10. $f(x) = x^3 - 4x^2 + x + 6$, degree 3. - Sum coef $1 - 4 + 1 + 6 = 4$. - Ends odd cubic. - Turning points 2. - Roots via factor or approx. - Y-int 6. 11. $f(x) = x^4 - 4x^3 - x^2 + 16x - 12$, degree 4. - Sum coef $1 - 4 -1 + 16 -12 = 0$. - Ends up. - Turning points max 3. - Roots via factorization. - Y-intercept -12. 12. $f(x) = x^2 + 3$, degree 2. - Sum coef $1 + 3 = 4$. - Ends up. - Turn 1. - X-intercepts none (no real root). - Y-int 3. 13. $f(x) = -x^4 + 3x^2 - x + 2$, degree 4. - Sum coef $-1 + 3 -1 + 2 = 3$. - Ends down. - Turn max 3. - Roots via approx. - Y-int 2. 14. $f(x) = 2x^5 - 2x^3 + 10x^2 - 13x + 3$, degree 5. - Sum coef $2 - 2 + 10 - 13 + 3 = 0$. - Ends: left down, right up. - Turn max 4. - Roots complex. - Y-int 3. 15. $f(x) = 2x(x - 5)(x - 2)$ degree 3. - Sum coef expand: $2x(x^2 - 7x + 10) = 2x^3 -14x^2 + 20x$ coef sum $2 - 14 + 20 = 8$. - Ends cubic odd. - Turning points 2. - X-int 0, 5, 2. - Y-int 0. Each graph shape corresponds to the degree and sign of leading term determining end behavior and number of turning points max degree-1. Intercepts found by roots and evaluating at zero.