Skip to content

Formulas: algebra and calculus, grade 10 (4 units)

Algebraic technique

  • The square of a sum
    (a+b)2\displaystyle (a+b)^2 =\displaystyle = a2+2ab+b2\displaystyle a^2+2ab+b^2
  • The square of a difference
    (a−b)2\displaystyle (a-b)^2 =\displaystyle = a2−2ab+b2\displaystyle a^2-2ab+b^2
  • The difference of squares
    (a+b)(a−b)\displaystyle (a+b)(a-b) =\displaystyle = a2−b2\displaystyle a^2-b^2
  • Factoring a trinomial – x1\displaystyle x_1, x2\displaystyle x_2 the roots of ax2+bx+c=0\displaystyle {ax^2+bx+c}={0}
    ax2+bx+c\displaystyle ax^2+bx+c =\displaystyle = a(x−x1)(x−x2)\displaystyle a(x-x_1)(x-x_2)
  • The roots of ax2+bx+c=0\displaystyle {ax^2+bx+c}={0}
    x1,2=−b±b2−4ac2a\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^2-4ac}}{2a}
  • The product of roots
    a⋅b\displaystyle \sqrt{a}\cdot\sqrt{b} =\displaystyle = ab\displaystyle \sqrt{ab}a≥0, b≥0\displaystyle a\geq0,\ b\geq0
  • The quotient of roots
    ab\displaystyle \frac{\sqrt{a}}{\sqrt{b}} =\displaystyle = ab\displaystyle \sqrt{\frac{a}{b}}a≥0, b>0\displaystyle a\geq0,\ b>0

The properties of a function

  • An even function – its graph symmetric about the y\displaystyle y-axis
    f(−x)\displaystyle f(-x) =\displaystyle = f(x)\displaystyle f(x)
  • An odd function – its graph symmetric about the origin
    f(−x)\displaystyle f(-x) =\displaystyle = −f(x)\displaystyle -f(x)
  • The x\displaystyle x-coordinate of the vertex of y=ax2+bx+c\displaystyle {y}={ax^2+bx+c}
    x=−b2a\displaystyle x=\frac{-b}{2a}
  • The absolute value
    ∣a∣={aa≥0−aa<0\displaystyle |a|=\begin{cases}a & a\geq0\\-a & a<0\end{cases}

Transforming a graph

  • A shift of k\displaystyle k units up (k<0\displaystyle k<0 – down)
    y=f(x)+k\displaystyle y=f(x)+k
  • A shift of p\displaystyle p units to the right (p<0\displaystyle p<0 – to the left)
    y=f(x−p)\displaystyle y=f(x-p)
  • A reflection in the x\displaystyle x-axis
    y=−f(x)\displaystyle y=-f(x)
  • A reflection in the y\displaystyle y-axis
    y=f(−x)\displaystyle y=f(-x)
  • A stretch (∣a∣>1\displaystyle |a|>1) or a shrink (∣a∣<1\displaystyle |a|<1)
    y=a⋅f(x)\displaystyle y=a\cdot f(x)

The derivative and the rules of differentiation

  • The average rate of change from x1\displaystyle x_1 to x2\displaystyle x_2 – the slope of the chord
    f(x2)−f(x1)x2−x1\displaystyle \frac{f(x_2)-f(x_1)}{x_2-x_1}
  • The derivative at a point – the slope of the tangent
    f′(x0)=lim⁡h→0f(x0+h)−f(x0)h\displaystyle f'(x_0)=\lim_{h\to0}\frac{f(x_0+h)-f(x_0)}{h}
  • The derivative of a power
    (xn)′\displaystyle (x^n)' =\displaystyle = nxn−1\displaystyle nx^{n-1}
  • The derivative of a constant
    (c)′\displaystyle (c)' =\displaystyle = 0\displaystyle 0
  • The derivative of a constant multiple
    (c⋅f(x))′\displaystyle (c\cdot f(x))' =\displaystyle = c⋅f′(x)\displaystyle c\cdot f'(x)
  • The derivative of a sum and of a difference
    (f(x)±g(x))′\displaystyle (f(x)\pm g(x))' =\displaystyle = f′(x)±g′(x)\displaystyle f'(x)\pm g'(x)
  • The derivative of a product
    (f(x)⋅g(x))′\displaystyle (f(x)\cdot g(x))' =\displaystyle = f′(x)⋅g(x)+f(x)⋅g′(x)\displaystyle f'(x)\cdot g(x)+f(x)\cdot g'(x)
  • The derivative of a composite function
    (f(g(x)))′\displaystyle (f(g(x)))' =\displaystyle = f′(g(x))⋅g′(x)\displaystyle f'(g(x))\cdot g'(x)

The tangent to a graph

  • The slope of the tangent at the point (x1,y1)\displaystyle (x_1,y_1) on the graph
    m=f′(x1)\displaystyle m=f'(x_1)
  • The equation of the tangent
    y−y1=m(x−x1)\displaystyle y-y_1=m(x-x_1)

The derivative of a square root function

  • The derivative of x\displaystyle \sqrt{x}
    (x)′\displaystyle (\sqrt{x})' =\displaystyle = 12x\displaystyle \frac{1}{2\sqrt{x}}x>0\displaystyle x>0
  • The derivative of the root of a function
    (f(x))′\displaystyle (\sqrt{f(x)})' =\displaystyle = f′(x)2f(x)\displaystyle \frac{f'(x)}{2\sqrt{f(x)}}f(x)>0\displaystyle f(x)>0

Rising, falling and extremum points

  • The function rises
    f′(x)>0\displaystyle f'(x)>0
  • The function falls
    f′(x)<0\displaystyle f'(x)<0
  • A point that may be an extremum
    f′(x)=0\displaystyle f'(x)=0