The companion set to Workshop 9. Twelve real SAT-style questions on quadratics, exponential growth, and functions. Read the form, multiply, use the machine.
The habit: read the form, multiply, use the machine.
Advanced Math rewards knowing what each form tells you. Quadratics: standard form shows the y-intercept, factored form shows the x-intercepts, vertex form shows the vertex, and the discriminant \(b^2 - 4ac\) counts the real solutions. Exponentials: a model \(a \cdot b^x\) starts at \(a\) and multiplies by \(b\) each step, so growth by 15% means \(\times 1.15\). Functions: \(f(x)\) is a machine; put the number in, work inside out, and remember that \(f(x) = 0\) marks the x-intercepts. Here is how the wrong answers try to trip you:
Sign Slip. Reads \((x - 9)\) as a zero at \(-9\), or flips a sign while substituting.
Wrong Form. Picks a form that hides the feature asked for (vertex form when the question wants the x-intercepts).
Boundary Value. Uses the value where the discriminant equals 0 (one solution) when the question asks for none.
Percent as the Base. Uses 1.5 for growth by 150%, or 0.15 for growth by 15%; the factor is 1 plus the rate.
Linear vs. Exponential. Adds a fixed amount when the quantity is multiplied by a factor, or the reverse.
Outside, Not Inside. Shifts right by changing the output instead of replacing \(x\) with \(x - h\).
Question 1 of 12
QuadraticsMedium
$(x + 2)(x - 9) = 0$
What is a positive solution to the given equation?
The takeaway: Set each factor equal to zero. From $x + 2 = 0$ you get $x = -2$, and from $x - 9 = 0$ you get $x = 9$. The only positive solution is $9$.
ExponentialsMedium
$p(x) = 1200(1.07)^x$
What is the value of $p(0)$?
The takeaway: Since $(1.07)^0 = 1$, we have $p(0) = 1200 \cdot 1 = 1200$. The value at $x = 0$ is the coefficient in front.
FunctionsMedium
The function $f$ is defined by $f(x) = x^2 - 3x + 1$.
The function $R$ is defined by
$R(x) = x^2 + 12x + 5$.
What is the value of $R(0)$?
The takeaway: Substituting $x = 0$ leaves only the constant: $R(0) = 5$. In standard form the constant $c$ is the y-intercept, so the graph crosses the y-axis at $(0, 5)$.
ExponentialsMedium
The expression $\dfrac{g^{4} \cdot g^{8}}{g^{5}}$ is equivalent to which of the following, where $g > 1$?
The takeaway: Add the numerator exponents on the common base: $g^{4} \cdot g^{8} = g^{12}$. Then subtract: $g^{12-5} = g^{7}$.
FunctionsMedium
The table gives four values of $x$ and the corresponding values of the polynomial function $f$.
x
f(x)
-2
20
0
-12
6
0
9
45
For which value of $x$ in the table is $(x, f(x))$ an x-intercept of the graph of $f$?
The takeaway: An x-intercept is a point where $f(x) = 0$. Among the listed inputs, only $x = 6$ has $f(x) = 0$, so $(6, 0)$ is an x-intercept.
QuadraticsChallenge
In the equation $x^2 + 14x + c = 0$, $c$ is a positive integer. The equation has no real solutions.
What is the least possible value of $c$?
The takeaway: No real solutions means the discriminant is negative: $b^2 - 4ac < 0$. Here $a = 1$, $b = 14$, so $196 - 4c < 0$, which gives $c > 49$. The least integer greater than $49$ is $50$.
ExponentialsChallenge
The value of quantity $P$ decreases by 3 percent of its original value each month. The value of quantity $Q$ decreases by 3 percent of its current value each month.
Which of the following statements is true?
The takeaway: For quantity $P$, 3 percent of the original value is the same fixed amount every month, so $P$ decreases by a constant amount, which is linear. For quantity $Q$, 3 percent of the current value shrinks as the value shrinks, so $Q$ is multiplied by $0.97$ each month, which is exponential decay.
FunctionsChallenge
The table gives three values of the function $f$.
x
f(x)
3
7
4
11
5
19
The function $g$ is defined by $g(x) = 2x - 1$. What is the value of $f(g(3))$?
The takeaway: Inner first: $g(3) = 2(3) - 1 = 5$. From the table, $f(5) = 19$.
QuadraticsChallenge
The function $g$ is defined by
$g(x) = -4x^2 + 16x + 84$.
Which of the following is an equivalent form of $g(x)$ that displays the $x$-intercepts of $y = g(x)$ as constants?
The takeaway: Factored form displays the $x$-intercepts: $-4x^2 + 16x + 84 = -4(x + 3)(x - 7)$, so the zeros are $x = -3$ and $x = 7$. Each form shows something different: standard form shows the y-intercept $(0, 84)$, and vertex form $-4(x - 2)^2 + 100$ shows the vertex $(2, 100)$, and because $a = -4$ is negative the parabola opens down, so $100$ is the maximum.
ExponentialsChallenge
A biologist counts a moss colony covering 24 square centimeters. The area increased by 150 percent each week.
Which equation gives the area $A$, in square centimeters, after $w$ weeks?
The takeaway: The initial area 24 is $a$. An increase by 150 percent makes the base $1 + 1.50 = 2.5$, so $A = 24(2.5)^w$.
FunctionsChallenge
The function $f$ is defined by
$f(x) = 2x^2$.
The function $g$ is the result of shifting $f$ to the right 6 units.
Which equation defines $g$?
The takeaway: A shift right 6 units replaces $x$ with $x - 6$: $g(x) = f(x - 6) = 2(x - 6)^2$. Right means minus inside. Shifts combine the same way: $g(x) = f(x - h) + k$ moves the graph right $h$ and up $k$. With a table: for $g(x) = f(x - 2) + 3$, finding $g(5)$ means reading $f(3)$ from the table and adding $3$.
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