August 2026 Algebra 1 Regents: Part I

Answer: (3), exponential growth.

1. Doubling means multiplying by 2 each day, rather than adding a fixed number.
2. If the initial population is \(P_0\), then after \(d\) days the model is \(P(d)=P_0(2)^d\). The variable appears in the exponent.
3. Since the multiplying factor is greater than 1, this is exponential growth, not decay.

SOMATH takeaway: A constant added amount suggests a linear model; a constant multiplying factor suggests an exponential model.

Answer: (1), \(30.5\).

1. Subtract \(2.6x\) from both sides: \(0.4x-4=8.2\).
2. Add 4: \(0.4x=12.2\). Divide by 0.4 to obtain \(x=30.5\).
3. Check: \(3(30.5)-4=87.5\), and \(2.6(30.5)+8.2=87.5\). Both sides agree.

SOMATH takeaway: Keep decimal place value straight: 12.2 divided by 0.4 is the same as 122 divided by 4.

Answer: (1), \(2(4y+12)+y=6\).

1. The second equation already isolates \(x\): \(x=4y+12\).
2. Replace the \(x\) in \(2x+y=6\) with the entire expression \(4y+12\). Parentheses show that the 2 multiplies all of it.
3. This gives \(2(4y+12)+y=6\). If instead you isolate \(y\), the correct expression is \(y=6-2x\), not either expression in choices (3) and (4).

SOMATH takeaway: Substitute an equal expression for a variable without changing its signs.

Answer: (2), \(80-n<100\).

1. “80 decreased by a number” means start at 80 and subtract the unknown number.
2. Write that expression as \(80-n\). “Is less than 100” supplies \(<100\).
3. The complete inequality is \(80-n<100\). Reversing the subtraction would describe a different statement.

SOMATH takeaway: Translate the order of subtraction before choosing the inequality sign.

Answer: (4), \(V(t)=29{,}500(1-0.175)^t\).

1. Convert the percent to a decimal: \(17.5\%=0.175\).
2. Each year the car keeps \(1-0.175=0.825\), or 82.5%, of its previous value.
3. Multiply the initial value by that remaining fraction once per year: \(V(t)=29{,}500(0.825)^t\). At \(t=0\), the formula gives the original $29,500.

SOMATH takeaway: For exponential decay, the base is the fraction remaining, not the percent lost.

Answer: (3), \((20-x)(20+x)\).

1. Recognize \(400=20^2\). Thus \(400-x^2=20^2-x^2\).
2. Use \(a^2-b^2=(a-b)(a+b)\) with \(a=20\) and \(b=x\).
3. The factors are \((20-x)(20+x)\). Choice (4) expands to \(x^2-400\), which has the opposite sign.

SOMATH takeaway: Identify the square roots of the two terms and preserve the order of subtraction.

Answer: (2), 10 devices.

1. Read household members on the horizontal axis and internet-connected devices on the vertical axis.
2. At six members, follow the overall upward pattern of the point cloud. A value near 10 devices falls through the middle of that pattern.
3. The question asks for a trend-based estimate, not an exact observed household at x = 6. Among the choices, 10 is the most reasonable estimate.

SOMATH takeaway: A scatter-plot prediction follows the overall association, not the highest or lowest point.

Answer: (4), \(x=3\).

1. For \(y=ax^2+bx+c\), the axis is \(x=-\frac{b}{2a}\). Here \(a=-1\) and \(b=6\).
2. Substitute: \(x=-6/[2(-1)]=3\).
3. The vertex is \((3,9)\), but the axis is the vertical line \(x=3\), not \(y=9\).

SOMATH takeaway: The axis of symmetry is an equation for a vertical line, not just the vertex height.

Answer: (2), addition property of equality.

1. Compare the two lines: the constant on the left rises from 3 to 15, and the right side rises from \(-12\) to 0.
2. Both sides have had 12 added: \(x^2-2x+3+12=-12+12\).
3. Adding the same number to both sides preserves equality. Reordering or regrouping terms is not the change shown here.

SOMATH takeaway: Name the operation actually performed on the equation.

Answer: (2), \(8x^4+5x^2+12\).

1. Fourth degree means the highest exponent is 4. A leading coefficient of 8 means the highest-degree term is \(8x^4\).
2. Standard form lists terms in descending order of exponent, ending with the constant 12.
3. Choice (2) meets all three requirements. Missing \(x^3\) and \(x\) terms simply have coefficient zero.

SOMATH takeaway: Check degree, leading coefficient, constant term, and ordering separately.

Answer: (4), 20.

1. Replace every \(x\) with \(-4\), using parentheses: \(g(-4)=(-4)^2-(-4)\).
2. The square is 16, and subtracting \(-4\) adds 4. Therefore \(g(-4)=16+4=20\).
3. Do not confuse \((-4)^2=16\) with \(-4^2=-16\). The substitution requires the parentheses.

SOMATH takeaway: Negative inputs need parentheses, especially when powers or subtraction are involved.

Answer: (4), 42 ounces.

1. The number of cookies triples: \(75/25=3\). Multiply the flour by the same factor.
2. The flour needed is \(3(1\frac34)=3(\frac74)=\frac{21}{4}=5.25\) cups.
3. Use the conversion supplied by the exam: \(5.25\times8=42\) ounces.

SOMATH takeaway: Scale the recipe and then convert the units; do not apply the scale factor twice.

Answer: (1), rational, because the product is 21.

1. Multiply: \((3\sqrt7)(\sqrt7)=3(\sqrt7\cdot\sqrt7)=3(7)=21\).
2. Since \(21=21/1\), it is rational.
3. The product of two irrational numbers is not always rational: for example, \(\sqrt2\sqrt3=\sqrt6\). So choice (2) reaches the right classification using a false general rule.

SOMATH takeaway: A multiple-choice justification must be true, not merely attached to the right conclusion.

Answer: (3), \(f(x)\) and \(g(x)\).

1. The formula \(f(x)=3(2)^x\) increases as \(x\) increases because the positive base 2 is greater than 1.
2. On the graph of \(g\), the entire interval from \(x=1\) to \(x=4\) lies on the rising right-hand branch.
3. For \(h\), the listed values from \(x=1\) through 4 are 4, 3, 4, 7. The initial drop from 4 to 3 means it is not increasing throughout the interval.

SOMATH takeaway: Check the whole stated interval; an overall rise between endpoints is not enough.

Answer: (4), \(\{0,-2,4\}\).

1. A zero is an input that makes the function equal 0. A product equals 0 when at least one factor equals 0.
2. Solve \(x=0\), \(x+2=0\), and \(x-4=0\). The results are \(0,-2,4\).
3. The initial factor \(x\) supplies the zero 0; do not overlook it.

SOMATH takeaway: Set each factor equal to zero, rather than copying the signs from inside the factors.

Answer: (2), I and III, only.

1. Rewrite III using base 2: \(16\cdot2^{x+2}=2^4\cdot2^{x+2}=2^{x+6}\). This is exactly I.
2. Rewrite II: \(4^{x+3}=(2^2)^{x+3}=2^{2x+6}\). Its exponent is not \(x+6\) for every \(x\).
3. At \(x=1\), I and III equal 128, but II equals 256. Agreement at a single input such as 0 would not prove equivalence.

SOMATH takeaway: Products with the same base add exponents; a power raised to a power multiplies exponents.

Answer: (3), \((x+2)^2=15\).

1. Move the constant: \(x^2+4x=11\).
2. Half of 4 is 2; its square is 4. Add 4 to both sides: \(x^2+4x+4=15\).
3. The left side factors as \((x+2)^2\), giving \((x+2)^2=15\).

SOMATH takeaway: Whatever you add to complete the square must also be added to the other side.

Answer: (1), George has the smaller standard deviation and the larger mean.

1. Enter each player's nine scores in a separate calculator list and run one-variable statistics.
2. George's mean is about \(171.78\), while Henry's is about \(170.11\). Thus George's mean is larger.
3. The population standard deviations are approximately \(38.63\) for George and \(42.67\) for Henry. Thus George's scores have the smaller spread. Using sample standard deviations consistently gives the same ordering.

SOMATH takeaway: The mean compares center; standard deviation compares spread. Compare both before choosing.

Answer: (1), each domain element maps to exactly one output.

1. Read the arrows: A maps to 1, B maps to 1, C maps to 3, and D maps to 2.
2. Each input has just one output, so the relation is a function.
3. Different inputs may share an output. A and B both mapping to 1 does not violate the function rule.

SOMATH takeaway: A function restricts outputs per input, not inputs per output.

Answer: (1), \(g(x)=|x+4|-5\).

1. A shift right by 2 replaces \(x\) with \(x-2\): \(g(x)=f(x-2)\).
2. Substitute: \(g(x)=|(x-2)+6|-5=|x+4|-5\).
3. Check the vertex: \((-6,-5)\) moves right to \((-4,-5)\). The vertical coordinate does not change.

SOMATH takeaway: For a horizontal shift, change the input inside the function.

Answer: (4), $0.11 per year.

1. Use the endpoint costs: $1.34 in 2009 and $2.08 in 2016.
2. Average rate of change is \(\frac{2.08-1.34}{2016-2009}=\frac{0.74}{7}\approx0.105714\).
3. Round to the nearest hundredth: $0.11 per year. There are seven elapsed years, even though the table lists eight calendar years.

SOMATH takeaway: Use change in output divided by change in input, not an average of the listed costs.

Answer: (4), \((3.5,-22.25)\).

1. Substitute the candidate x-coordinate 3.5: \(y=(3.5)^2-7(3.5)-10\).
2. Compute carefully: \(12.25-24.5-10=-22.25\).
3. Thus \((3.5,-22.25)\) lies on the graph. For comparison, \(x=10\) gives 20, and \(x=-10\) gives 160, so neither point with y-coordinate 0 works.

SOMATH takeaway: A point belongs to a graph only when its two coordinates satisfy the equation together.

Answer: (3), \(x^2(x-9)(x+4)\).

1. First factor out the greatest common factor \(x^2\): \(x^2(x^2-5x-36)\).
2. Find two numbers with product \(-36\) and sum \(-5\): \(-9\) and 4.
3. The trinomial becomes \((x-9)(x+4)\). Keep the GCF to obtain \(x^2(x-9)(x+4)\).

SOMATH takeaway: Factoring out the GCF is the first step, not necessarily the final one.

Answer: (4), \(x^2+7x-7\).

1. “A is subtracted from B” means \(B-A\). Write \((4x^2+5x-6)-(3x^2-2x+1)\).
2. Distribute the subtraction to every term: \(4x^2+5x-6-3x^2+2x-1\).
3. Combine like terms: \(x^2+7x-7\). In particular, \(5x-(-2x)=7x\).

SOMATH takeaway: Write parentheses before subtracting, then change every sign in the subtracted polynomial.