Digital SAT Math: Algebra, Advanced Math & Geometry · Public flashcards

Question 1: When does a system of two linear equations in two variables have infinitely many solutions, and when does it have no solution?

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Infinitely many solutions occur when the lines are identical (same slope $m_1 = m_2$ and same y-intercept $b_1 = b_2$ ). No solution occurs when the lines are parallel and distinct (same slope $m_1 = m_2$ , but different y-intercepts $b_1 \ne b_2$ ).

In standard form $A_1x + B_1y = C_1$ and $A_2x + B_2y = C_2$ , infinitely many solutions mean $\frac{A_1}{A_2} = \frac{B_1}{B_2} = \frac{C_1}{C_2}$ . No solution means $\frac{A_1}{A_2} = \frac{B_1}{B_2} \ne \frac{C_1}{C_2}$ .

Question 2: How can the built-in Desmos graphing calculator be used to instantly solve a system of linear or non-linear equations?

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Enter both equations exactly as written (e.g., $y = 3x - 5$ and $x^2 + y^2 = 25$ ) on separate lines; the coordinates $(x, y)$ of their intersection points are displayed as clickable gray dots.

Desmos handles equations without needing to isolate $y$ , eliminating algebraic manipulation errors under time pressure.

Question 3: What are the coordinates of the vertex of a parabola in standard form $y = ax^2 + bx + c$ , and what is the axis of symmetry?

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The vertex is at $(h, k)$ where $h = -\frac{b}{2a}$ and $k = f\left(-\frac{b}{2a}\right)$ . The axis of symmetry is the vertical line $x = -\frac{b}{2a}$ .

If $a > 0$ , the parabola opens upward and $k$ is the minimum value; if $a < 0$ , the parabola opens downward and $k$ is the maximum value.

Question 4: What are Vieta's formulas for the sum and product of the roots of a quadratic equation $ax^2 + bx + c = 0$ ?

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Sum of the roots: $r_1 + r_2 = -\frac{b}{a}$ . Product of the roots: $r_1 \cdot r_2 = \frac{c}{a}$ .

Vieta's formulas bypass factoring or the quadratic formula when the question asks for the sum or product of solutions, saving significant time.

Question 5: How do you find the number of real solutions of $ax^2 + bx + c = 0$ using the discriminant $D = b^2 - 4ac$ ?

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If $D > 0$ , there are 2 distinct real solutions; if $D = 0$ , there is exactly 1 real solution (tangent to x-axis); if $D < 0$ , there are no real solutions.

On the Digital SAT, questions asking for what value of $k$ the equation has exactly one solution frequently require setting $b^2 - 4ac = 0$ and solving for $k$ .

Question 6: What is the general model for exponential growth and exponential decay, and what do the parameters represent?

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$f(t) = a(1 + r)^t = a \cdot b^t$ for growth ( $b > 1$ , where $r$ is the percent growth rate as a decimal), and $f(t) = a(1 - r)^t = a \cdot b^t$ for decay ( $0 < b < 1$ ).

The parameter $a$ is the initial value at $t = 0$ (the y-intercept), and $b$ is the growth or decay factor per unit time.

Question 7: How does changing the time exponent to $\frac{t}{k}$ affect an exponential model $f(t) = a \cdot b^{t/k}$ ?

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The value multiplies by factor $b$ every $k$ units of time (e.g., $f(t) = 100(2)^{t/3}$ doubles every 3 hours).

Digital SAT questions frequently ask to convert annual growth rates to monthly rates: $(1 + r)^t = ((1 + r)^{1/12})^{12t}$ , where the monthly growth factor is $(1 + r)^{1/12}$ .

Question 8: What is the standard equation of a circle in the xy-plane, and what do $(h, k)$ and $r$ represent?

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$(x - h)^2 + (y - k)^2 = r^2$ , where $(h, k)$ is the center of the circle and $r$ is the radius ( $r > 0$ ).

When given in general form $x^2 + y^2 + Dx + Ey + F = 0$ , complete the square for both $x$ and $y$ terms to find the center and radius.

Question 9: How can you instantly graph circles and find their center and radius using Desmos on the SAT?

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Type the circle equation directly into Desmos even in expanded form (e.g., $x^2 + y^2 - 6x + 8y = 0$ ). Desmos automatically graphs the circle; inspect the center by observing coordinates or plotting the midpoint of horizontal/vertical diameter extrema.

Desmos eliminates algebraic errors from completing the square under timed conditions.

Question 10: What is the relationship between an inscribed angle and the central angle that subtends the same arc in a circle?

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The inscribed angle measures exactly half of the central angle (and half of the intercepted arc measure): $\theta_{\text{inscribed}} = \frac{1}{2} \theta_{\text{central}}$ .

Consequently, any inscribed angle that intercepts a diameter (a semicircle of $180^\circ$ ) is a right angle ( $90^\circ$ , Thales's Theorem).

Question 11: How are arc length $s$ and sector area $A_{\text{sector}}$ calculated for a central angle of $\theta$ radians and radius $r$ ?

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Arc length $s = r\theta$ ; Sector area $A_{\text{sector}} = \frac{1}{2} r^2 \theta$ .

If the central angle is in degrees: $s = 2\pi r \left(\frac{\theta}{360^\circ}\right)$ and $A_{\text{sector}} = \pi r^2 \left(\frac{\theta}{360^\circ}\right)$ .

Question 12: What is the fundamental cofunction identity relating sine and cosine for complementary acute angles?

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$\sin(x) = \cos(90^\circ - x)$ , or in radians $\sin(x) = \cos\left(\frac{\pi}{2} - x\right)$ .

A Digital SAT staple: if $\sin(a) = \cos(b)$ for acute angles $a$ and $b$ in a right triangle, then $a + b = 90^\circ$ (or $\frac{\pi}{2}$ ).

Question 13: What are the side length ratios for the $30^\circ$ - $60^\circ$ - $90^\circ$ and $45^\circ$ - $45^\circ$ - $90^\circ$ special right triangles?

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For $30^\circ$ - $60^\circ$ - $90^\circ$ : $x : x\sqrt{3} : 2x$ (opposite $30^\circ$ is $x$ , opposite $60^\circ$ is $x\sqrt{3}$ , hypotenuse is $2x$ ). For $45^\circ$ - $45^\circ$ - $90^\circ$ : $x : x : x\sqrt{2}$ .

These ratios are also provided on the reference sheet at the beginning of each SAT math module, but memorizing them saves valuable test time.

Question 14: How do you find the slope of a line perpendicular to a given line with slope $m$ ?

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The perpendicular slope is the negative reciprocal: $m_{\perp} = -\frac{1}{m}$ (assuming $m \ne 0$ ).

The product of the slopes of two perpendicular non-vertical lines is always $m_1 \cdot m_2 = -1$ . Horizontal lines ( $m = 0$ ) are perpendicular to vertical lines (undefined slope).

Question 15: How does the built-in Desmos regression feature find unknown coefficients from a table or equation?

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Use the tilde `~` operator instead of `=` (e.g., $y_1 \sim a x_1^2 + b x_1 + c$ ). Desmos automatically computes and displays the values of $a$ , $b$ , and $c$ along with $R^2$ .

This shortcut is especially powerful for problems with several given data points or graphs where you need to deduce an exact polynomial or exponential model.

Question 16: What is the difference between standard deviation and range as measures of data spread?

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Range is simply $\text{Maximum} - \text{Minimum}$ , sensitive only to extreme values. Standard deviation measures how spread out individual data points are relative to the mean.

On the SAT, you rarely need to compute numerical standard deviation; you only need to recognize that data clustered closer to the mean has a smaller standard deviation than widely dispersed data.

Question 17: How does adding a constant $c$ to every value in a dataset affect the mean, median, standard deviation, and range?

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The mean and median both increase by $c$ ( $\text{new} = \text{old} + c$ ), while the standard deviation and range remain completely unchanged.

Shifting all data points translates the distribution along the axis without altering its dispersion or shape.

Question 18: How is conditional probability calculated from a two-way contingency table on the SAT?

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$P(A|B) = \frac{\text{Number in category } A \text{ and } B}{\text{Total in condition category } B}$ . Restrict your denominator strictly to the given condition group, not the grand total.

Phrases like 'Given that the person is in group B...' or 'Of those who tested positive...' signal that the total sample space is restricted to group B.

Question 19: What is the relationship between the linear scale factor $k$ , the area ratio, and the volume ratio for similar geometric solids?

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If the ratio of corresponding linear dimensions is $k = \frac{L_1}{L_2}$ , the ratio of surface areas is $k^2 = \frac{A_1}{A_2}$ and the ratio of volumes is $k^3 = \frac{V_1}{V_2}$ .

If a solid has its linear dimensions tripled ( $k = 3$ ), its surface area increases by $3^2 = 9$ times, and its volume increases by $3^3 = 27$ times.

Question 20: What are the coordinates of the midpoint of a line segment connecting $(x_1, y_1)$ and $(x_2, y_2)$ , and what is the distance formula?

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Midpoint $M = \left(\frac{x_1 + x_2}{2}, \frac{y_1 + y_2}{2}\right)$ . Distance $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$ .

The distance formula is a direct application of the Pythagorean Theorem $a^2 + b^2 = c^2$ in the Cartesian coordinate plane.