2026 Math SAT Study Guide – Part 1

NONLINEAR FUNCTIONS

• The half life of Carbon${}_{14}$ is 5,730 years. The equation for how much Carbon${}_14$ of a 400 gram sample is left after t years $f(t)=400 {\left(\dfrac{1}{2}\right)}^{t/5730}$. $f(7000)$ would be the amount of the sample left after 7,000 years.
• If the function $f(x)=x^3+11$ is translated 3 down and 2 to the left, the new function $g(x)$ is $g(x)={(x + 2)}^2+11 -3={(x+2)}^2+8$.
• A projectile is fired from ground level at $t=0$, reached a maximum height of 73 feet at $t=6$ and hits the ground at $t=12$. An equation for its height is $h(t)=a{(x -6)}^2+77$, by the formula for a parabola from its vertex. Now we need to determine $a$. Plugging in the initial values of $(0,0)$, $0=a{(0 -6)}^2+77\implies 0=36a+77\implies a=\dfrac{-77}{36}$. So the equation is $h(t)= \dfrac{-77}{36}{(x -6)}^2+77$.
• If $h(x)= -16x^2+40x+10$ is the height above ground of a projectile. The meaning of the positive $x-$intercept is the time when the projectile hit the ground. $-16$ is downward acceleration due to gravity, 40 is the speed it was shot upward, and 10 is its initial height.
• If $f(x)=(x -7)(x+11)$, the $x-$intercepts are $-11$ and 7. The $x-$coordinate of the vertex is the average of the intercepts $\dfrac{7 - 11}{2}= -2$. You can also find the vertex by FOILing and using the vertex formula $\dfrac{-b}{2a}$ or by graphing.
• If $f(x)=7\cdot 2^x$, $f(x+3)=7\cdot 2^{x+3}=7\cdot 2^x\cdot 2^3=7\cdot 2^x\cdot 8=56\cdot 2^x.$
• $f(x)=(x+3)(x -2)(x -5)$ has $x-$intercepts at $x= -3$, 2, and 5. $f(x)+7$ has $y-$values of 7 at $-3$, 2, and 5.
• It is important to understand that if there were $12,000$ people and the population increased by $15\%$, the new population would be $12,000\cdot 1.15$. The existing amount is 1 and it is increased by $15\%$ meaning 0.15, so $1+0.15=1.15$.
• Similarly, if you bought a car for $\$50,000$ and its value depreciated by $15\%$ per year, you multiply by $1 -0.15 = 0.85$. So the car’s value after $t$ years would be $50,000\cdot .85^t$.
• If you are given the zeros of a third degree equation are 2, 3, an 5 and asked the find $c$ in $y=x^3+ax^2+bx+c$, the factors are $(x -2)(x -3)(x -5)$. Multiplying the constant terms $(-2)\cdot (-3)\cdot (-5)= -30$, so $c= -30$. It is not necessary to find the whole polynomial.
• $f(x)=(x+2)(x -4)(x+5)$ is shifted down 7 units and the resulting graph goes through $(3,j)$. What is $j$? $g(x)=(x+2)(x -4)(x+5) -7$. Now find $g(3)$. $g(3)=(3+2)(3 -5)(3+5) -7=5\cdot (-2)\cdot 8 -7= -87$.
• What is a quadratic equation with solutions $x=2$ and $x= -5$? $x=2\implies x -2=0$, $x= -5\implies x+5=0$. $(x -2)(x+5)=0\implies x^2+5x -2x -10=0\implies x^2+3x -10=0$.
• A town has a population of $30,000$ and is increasing at $7\%$ per year. At this rate, how many years will it take to get to $70,000$? $30,000{(1.07)}^t=70,000$. You should not have to actually solve for $t$.
• Which quadratic equation goes through the following points $( -1,3)$, $(0,5)$ and $(1,11)$? It may be easier to plug the points into the answer choices. Substitute into $ax^2+bx+c=y$ and get 3 equations and 3 variables. $(-1)^2a+( -1)b+c=3\implies a -b+c=3$. $0^2a+0b+c=5\implies c=5$. $1^2a+b1+c=11\implies a+b+c=11$. $a -b+5=3\implies a -b= -2$. $a+b+5=11\implies a+b=6$. Adding equations $2a=4\implies a=2$. $2+b=6\implies b=4$. So $y=2x^2+4x+5$.
• For what value of $x$ does $y=(x -11)(x+24)$ reach its minimum? You can graph it and find the minimum / vertex. The $x-$intercepts are 11 and $-24$. Their average is $\dfrac{-13}{2}$. FOILing $x^2+13x -264=y$. Using the vertex formula $\dfrac{-b}{2a}=\dfrac{-13}{2}$.
• $y=a^x+b$ goes through $(0,3)$ and $(3,29)$. What are $a$ and $b$? $3=a^0+b\implies 3=1+b\implies b=2$, so $y=a^x+2$. Now substitute $(3,29)$, $29=a^3+2\implies 27=a^3\implies a=3$. So $y=a^3+2$.
• In $P=P_0(1+r)^t$ modeling population, if $r>0$, the population is increasing, but if $-1 < r < 0$, the population is decreasing.
• For what value of $x$ does $f(x)=(x -17)^2+11$ reach a minimum? The equation is in vertex for $y(x -a)^2+b$, so 17. It is also possible to graph it or FOIL it out and use the vertex formula $\dfrac{-b}{2a}$.
• What is the $y-$intercept of $y= -11(7)^x+18$? $-11(7)^0+18= -11+18=7$.
• If there is a factor of $(x -a)$, there is an $x-$intercept of $a$. If there is a factor of $(x -a)^2$, the graph touches at $a$ and goes back in the same direction. With a single $(x -a)$ factor, the graph goes through at $a$.
• Below is the graph of $y=(x -2)(x -5)^2$.

• If $f(t)=200\cdot 2^{t/150}$ gives the number of bacteria in a sample, the number of bacteria doubles every 150 minutes, when you get $2^{150/150}=2^1=2$. It also doubles every $\dfrac{150}{60}=2\dfrac{1}{2}$ hours.
• If $f(2)=k$, which shows $k$ as a constant or coefficient $100\cdot 2(x-2)$, because at 2, you get $100\cdot 2(2 -2)=100\cdot 2^0=100\cdot 1=100$.

EXPRESSIONS

• Which is equivalent to ${\left(x +\dfrac{y}{4}\right)}^2$? It is a trap to square the terms in place rather than FOIL. $\left(x+\dfrac{y}{4}\right)\left(x+\dfrac{y}{4}\right)=x^2+\dfrac{xy}{4}+\dfrac{xy}{4}+\dfrac{y^2}{16}=x^2+\dfrac{xy}{2}+\dfrac{y^2}{16}$.
• Which is a factor of $9x^2+24xy+16y^2$. $3x+4y$. It is best to recognize the perfect square formula $(a+b)^2=a^2+2ab+b^2$. Notice that the first and last terms are perfect squares and take the square root of each of them. Dividing the answer choices works, but is difficult and time consuming. It is possible to factor or use the quadratic formula even though there are two variables.
• If $4^{5a}=\sqrt[7]{2^3}$, what is $a$? ${2^2}^{5a}={\left(2^3\right)}^{1/7}\implies 2^{10a}=2^{3/7}\implies 10a=\dfrac{3}{7}\implies a=\dfrac{3}{70}$.
• $(ax+5)(2x^2+bx+7)=8x^3+22x^2+43x+35$. If this expression is true for all $x$, what is $a\cdot b$? FOIL $2ax^3+abx^2+7ax+10x^2+5bx+35$. Equating coefficients of like terms, $2a=8\implies a=4$, $ab+10=22$, $7a+5b=43$. Substituting $4b+10=22\implies b=3$. $a\cdot b=12$.
• Which is equivalent to $x^2 -3$? This is a difference of squares $x^2 -\left({\sqrt{3}}\right)^2=\left(x+\sqrt{3}\right)\left(x -\sqrt{3}\right)$. It is also possible to plug in a value for $x$, to multiply out the answer choices, or to graph the original expression and the answer choices, but all of those would be more time consuming.
• $\dfrac{x^2}{5} -3=\dfrac{1}{5}\left(x -\sqrt{k}\right)\left(x+\sqrt{k}\right)$. What is $k$? $\dfrac{1}{5}\left(x^2 -15\right)=\dfrac{1}{5}\left(x+\sqrt{15}\right)\left(x -\sqrt{15}\right)$. $k=15$.

PERCENTAGES

Percent change: $\dfrac{\textup{new } - \textup{ original}}{\textup{original}}\cdot 100$. For example, an increase from 4 to 5 is $\dfrac{5 -4}{4}\cdot 100=25\%$. A decrease from 5 to 4 is $\dfrac{4 -5}{5}\cdot 100= -20\%$ or a $20\%$ decrease.

• An item costs $\$80$ after a $7\%$ sales tax. What was the before tax price of the item? $80=1.07x\implies \dfrac{80}{1.07}=x\implies x\approx \$74.77$.
• An item costs $\$80$ after a $30\%$ discount. What was the before discount price of the item? $(1 -0.3)x=80\implies 0.7x=80\implies x=\dfrac{80}{0.7}\approx \$114.29$.
• Depreciated value $=A=P(1 -r)^t$. You buy a computer for $\$800$ and its value depreciates at $20\%$ per year. How much will it be worth after 2 years? $800(.75)^2=\$450$.
• $0.34x$ represents a decrease in $x$ by what percent of $x$? $(1 -0.34)\cdot 100=66\%$.

TRIGONOMETRY

The Pythagorean theorem: $a^2+b^2=c^2$, where $a$ and $b$ are legs of a right triangle and $c$ is the hypotenuse.

• If the hypotenuse of a right triangle is 8 and one side is $2\sqrt{5}$, what is the length of the other leg? $\left(2\sqrt{5}\right)^2+b^2=8^2\implies 4\cdot 5+b^2=64\implies b^2=44\implies b=2\sqrt{11}$.
• In a $30^{\circ}-60^{\circ}-90^{\circ}$ triangle, the side are in the ratios of $1-\sqrt{3}-2$.
• If a square is inscribed in a circle, the side lengths of the square are $\sqrt{2}$ times the radius or $\dfrac{\sqrt{2}}{2}$ times the diameter. The sides of the square are the legs of an isosceles right triangle with the diameter being the hypotenuse.

AREA AND VOLUME

• What is the perimeter of a square with the same area as a circle with radius 6. Area of square $=s^2$. $6=s^2\implies s=6$. Perimeter $=4s=4\cdot 6=24$.
• Problems with cylinders, cones, etc. may give the diameter and height and ask you to find the volume. It is important to realize that the formulas are given in terms of the radius. You need to divide the diameter by 2 to get the radius. These problems are not hard, but it is a trap to use diameter as radius or not to be able to compute the radius.
• The volume of a cylinder is $6\pi$ cubic feet. Its height is 12 feet. What is its radius. $\pi r^2h=$ Volume. $\pi r^2\cdot 12=6\pi\implies r^2=\dfrac{1}{2}=\dfrac{\sqrt{2}}{2}$.
• The ratio of the area of similar $2-$dimensional figures is proportional to the square of their linear measures. The ratio of the volume of similar $3-$dimensional figures is proportional to the cube of their linear measures.

LINEAR FUNCTIONS

• $f(32)=43$ and $f(37)=30$ and $f$ is a line, what is the equation of $f(x)$? $m=\dfrac{y2 -y1}{x2 -x1}=\dfrac{30 -43}{37 -32}=\dfrac{-13}{5}$. $y=mx+b$. $43=\dfrac{-13}{5}\cdot 32+b\implies 43=-\dfrac{416}{5}+b\implies b=43+\dfrac{416}{5}=\dfrac{215}{45}+\dfrac{416}{5}=\dfrac{631}{5}$. So $y=\dfrac{-13x}{5}+\dfrac{631}{5}$.
• $f(x)=mx+187$ and $f(12)=55$, what is $f(15)$? $55=m\cdot 12+187\implies -132=12m\implies m=-11$. So $f(x)= -11x+187$. $f(15)= -11\cdot 15+187= -165+187=22$.
• If $y=3x+21$, to find the $y-$intercept, set $x$ equal to 0. $y=0x+21\implies y=21$, so $(0,21)$. To find the $x-$intercept, set $y$ equal to 0 and solve for $x$: $0=3x+21\implies 3x= -21\implies x= -7$, so $(-7,0)$.
• It is important in many problems to be able to find the equation of a line given 2 points. First find the slope. Then use $y=mx+b$ to solve for $b$ or $y -y_0=m(x -x_0)$ to find the equation of the line.
• Find the equation of the line through $(2,3)$ and $(5,15)$. $m=\dfrac{y2 -y1}{x2 -x1}$. $m=\dfrac{15 -3}{5 -2}=\dfrac{12}{3}=4$. Now take one of the points into $y=mx+b$. $3=4\cdot 2+b\implies b= -5$, so $y=4x -5$. Or $y -3=4(x -2)\implies y -3=4x -8\implies y=4x -5$.
• If $f(4x)=2x -5$, What $f(12)$? $f(12) = f(4\cdot 3)$, so $x=3$. Now substitute 3 for $x$. $f(12)=2\cdot 3 -5=1$.
• If $y=3x+7$, what is the sum of the $x-$intercept and the $y-$intercept. The $y-$intercept $y=3\cdot 0+7\implies y=7$, so $(0,7)$. $x-$intercept, $0=3x+7\implies -7=3x\implies x=\dfrac{-7}{3}$, so $\left(\dfrac{-7}{3}, 0\right)$. $7+\left(\dfrac{-7}{3}\right)=\dfrac{21}{3} -\dfrac{7}{3}=\dfrac{14}{3}$.