Quadratic and Polynomial Relationships - Quantitative Reasoning

1. What is the vertex of $y=ax^2+bx+c$ written in terms of $a,b,c$?

   $\left(-\frac{b}{2a},\frac{4ac-b^2}{4a}\right)$. The vertex coordinates are found by using the axis of symmetry for $x$ and substituting to find $y$, yielding the given expressions in terms of $a$, $b$, $c$.

2. What is the axis of symmetry of $y=ax^2+bx+c$?

   $x=-\frac{b}{2a}$. The axis of symmetry for a parabola $y = ax^2 + bx + c$ is the vertical line through the vertex at $x = -b/(2a)$.

3. Solve $x^2-5x+6=0$.

   $x=2,3$. Factoring gives $(x-2)(x-3)=0$, so roots are $2$ and $3$.

4. What is the relationship between roots and coefficients for a monic quadratic $x^2+px+q$?

   Roots sum $=-p$, product $=q$. For monic quadratics, Vieta's formulas simplify to sum of roots $=-p$ and product $=q$.

5. What is the factored form of a quadratic with roots $r_1$ and $r_2$?

   $a(x-r_1)(x-r_2)$. The factored form arises from the roots being the values that make the quadratic zero, scaled by the leading coefficient $a$.

6. What is the quadratic formula for solutions of $ax^2+bx+c=0$?

   $x=\frac{-b\pm\sqrt{b^2-4ac}}{2a}$. The quadratic formula derives from completing the square to solve $ax^2 + bx + c = 0$ for $x$.

7. What is the discriminant of $ax^2+bx+c$ and what does its sign determine?

   $\Delta=b^2-4ac$; sign gives number of real roots. The discriminant $Delta = b^2 - 4ac$ determines the nature of roots: positive for two real, zero for one real, negative for no real roots.

8. Find the quadratic with roots $2$ and $-3$ and leading coefficient $1$.

   $x^2+x-6$. Using sum $2 + (-3) = -1$ and product $2 \times (-3) = -6$ in monic form.

9. What is the axis of symmetry of $y=5x^2-20x+1$?

   $x=2$. The axis of symmetry is $x = -b/(2a) = 20/(10) = 2$.

10. Factor $x^2-9x+20$ over the integers.

    $(x-5)(x-4)$. Factors are found by pairs that multiply to $20$ and add to $-9$, yielding $ -4$ and $-5$.

11. Identify the discriminant of $2x^2+3x-7$.

    $\Delta=65$. The discriminant is calculated as $b^2 - 4ac = 9 - 4(2)(-7) = 9 + 56 = 65$.

12. Find the sum and product of roots of $3x^2-12x+5=0$.

    Sum $=4$, product $=\frac{5}{3}$. Applying Vieta's formulas to $3x^2 - 12x + 5 = 0$ gives sum $=12/3=4$ and product $=5/3$.

13. What are the sum and product of roots of $ax^2+bx+c=0$ in terms of $a,b,c$?

    Sum $=-\frac{b}{a}$, product $=\frac{c}{a}$. Vieta's formulas state that for $ax^2 + bx + c = 0$, the sum of the roots is $-b/a$ and the product is $c/a$.

14. Identify the number of real roots of $ax^2+bx+c=0$ when $b^2-4ac>0$.

    Two distinct real roots. A positive discriminant indicates the quadratic crosses the x-axis at two distinct points.

15. What is the vertex form of a quadratic, and what is the vertex in that form?

    $y=a(x-h)^2+k$; vertex $=(h,k)$. Vertex form is obtained by completing the square, with $(h, k)$ representing the vertex's coordinates.

16. What is the minimum or maximum value of $y=a(x-h)^2+k$ in terms of $a$ and $k$?

    If $a>0$, min $=k$; if $a<0$, max $=k$. In vertex form, $k$ is the extremum value, minimum if $a>0$ and maximum if $a<0$.

17. If roots of $x^2-kx+12=0$ are $3$ and $4$, what is $k$?

    $k=7$. Sum of roots $3+4=7$ equals $k$ for the quadratic $x^2 - kx +12=0$.

18. Identify the number of real roots of $ax^2+bx+c=0$ when $b^2-4ac=0$.

    One real double root. A zero discriminant means the quadratic touches the x-axis at exactly one point, a repeated real root.

19. What is the $y$-intercept of $y=ax^2+bx+c$?

    $c$. The $y$-intercept occurs at $x=0$, so substituting gives $y=c$.

20. What is the quadratic whose roots are $r$ and $s$ and whose leading coefficient is $1$?

    $x^2-(r+s)x+rs$. The monic quadratic is constructed using Vieta's formulas with sum $r+s$ and product $rs$.

21. Identify the number of real roots of $ax^2+bx+c=0$ when $b^2-4ac<0$.

    No real roots (two complex conjugates). A negative discriminant indicates no real solutions, resulting in two complex conjugate roots.

22. What is the condition on $a$ for $y=ax^2+bx+c$ to open upward versus downward?

    Upward if $a>0$; downward if $a<0$. The sign of $a$ determines the parabola's direction: positive $a$ opens upward, negative $a$ opens downward.

23. Find the vertex of $y=x^2-6x+11$.

    $(3,2)$. The vertex is at $x=3$, and $y= (3)^2 -6(3) +11=2$, so $(3,2)$.

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