Passport to Advanced Math

Passport to Advanced Math — SAT Topic Page

Exam-oriented practice on nonlinear functions and expressions: quadratics, exponentials, radicals, rational expressions, and function structure.

Quick Reference (Minimal Theory)

Concept	Form / Cue	Tip
Quadratic vertex	\(x_v=-\tfrac{b}{2a}\), \(y_v=f(x_v)\)	Complete square
Discriminant	\(\Delta=b^2-4ac\)	Root count
Exponential	\(y=ab^x\)	Growth vs decay
Rational domain	denominator \(\ne0\)	Asymptotes
Inverse function	swap \(x,y\), solve	Reflect

Worked Examples (10) — With Graphs

Example 1 Ivy

Find the vertex and intercepts of \( f(x)=x^2-6x+5 \); sketch behavior.

Show solution

\(x_v=\tfrac{6}{2}=3\), \(f(3)=9-18+5=-4\) → vertex \((3,-4)\). Zeros from factoring: \((x-1)(x-5)=0\Rightarrow x=1,5\). Y-int at \(x=0\Rightarrow 5\).

Vertex \((3,-4)\); x-ints 1 and 5; y-int 5.

Example 2 MIT

Given \( g(x)=2x^2-8x+k \) has exactly one real root. Find \(k\).

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One real root ⇒ \(\Delta=0\). \(b^2-4ac=(-8)^2-4(2)(k)=64-8k=0\Rightarrow k=8\).

\(k=8\).

Example 3 Ivy

An exponential model \( P(t)=P_0(1.12)^t \). If \(P_0=500\), find \(P(5)\) and the doubling time (approx).

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\(P(5)=500\cdot1.12^5\approx500\cdot1.762=881\). Doubling time \(t=\log 2/\log 1.12\approx 6.12\).

\(P(5)\approx881\); doubling \(\approx6.1\) periods.

Example 4 MIT

Solve \( \sqrt{3x+1}=x-1 \) for real \(x\).

Show solution

Require \(x\ge1\). Square: \(3x+1=x^2-2x+1\Rightarrow x^2-5x=0\Rightarrow x(x-5)=0\Rightarrow x=0,5\). Only \(x=5\) meets \(x\ge1\). Check: \(\sqrt{16}=4=5-1\).

\(x=5\).

Example 5 Ivy

Simplify and state domain: \( \dfrac{x^2-9}{x^2-5x+6} \).

Show solution

Factor: numerator \((x-3)(x%2B3)\); denominator \((x-2)(x-3)\). Simplifies to \(\dfrac{x%2B3}{x-2}\) with restriction \(x\ne2,3\).

\(\dfrac{x%2B3}{x-2}\); domain \(x\ne2,3\).

Example 6 MIT

If \( f(x)=\dfrac{ax%2Bb}{x-1} \) and \(f(0)=2\), \(f(2)=4\), find \(a,b\).

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\(f(0)=\tfrac{b}{-1}=2\Rightarrow b=-2\). \(f(2)=\tfrac{2a-2}{1}=4\Rightarrow 2a-2=4\Rightarrow a=3\).

\(a=3,\; b=-2\).

Example 7 Ivy

Find the inverse of \( h(x)=\dfrac{3x-2}{x%2B4} \); state any restrictions.

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Let \(y=\dfrac{3x-2}{x%2B4}\Rightarrow y(x%2B4)=3x-2\Rightarrow yx%2B4y=3x-2\Rightarrow (3-y)x=4y%2B2\Rightarrow x=\dfrac{4y%2B2}{3-y}\). Swap: \(h^{-1}(x)=\dfrac{4x%2B2}{3-x}\); restrict \(x\ne3\) and original domain \(x\ne-4\).

\(h^{-1}(x)=\dfrac{4x%2B2}{3-x}\); domain excludes \(x=3\).

Example 8 MIT

Given \( f(x)=x^2-4x \) and \( g(x)=\sqrt{x%2B1} \), find \( (g\circ f)(x) \) and its domain.

Show solution

\((g\circ f)(x)=\sqrt{f(x)%2B1}=\sqrt{x^2-4x%2B1}=\sqrt{(x-2)^2-3}\). Need radicand \(\ge0\Rightarrow (x-2)^2\ge3\Rightarrow x\le 2-\sqrt3\) or \(x\ge 2%2B\sqrt3\).

\( (g\circ f)(x)=\sqrt{x^2-4x%2B1} \); domain \(x\le 2-\sqrt3\) or \(x\ge2%2B\sqrt3\).

Example 9 Ivy

Polynomial \( p(x)=x^3-4x^2+ax%2B6 \) has \(x=1\) as a root. Factor \(p(x)\) and find \(a\) if another root is 2.

Show solution

Factor by root 1: synthetic division gives \(p(x)=(x-1)(x^2-3x%2B(a-6))\). If 2 is a root, plug: \(8-16%2B2a%2B6=0\Rightarrow -2%2B2a=0\Rightarrow a=1\). Then quad \(x^2-3x-5\).

\(p(x)=(x-1)(x^2-3x-5)\); \(a=1\).

Example 10 MIT

Find all real \(x\) such that \( \dfrac{x-1}{x%2B2} + \dfrac{x%2B1}{x-2} = 1 \).

Show solution

LCD \((x%2B2)(x-2)\). Numerator: \((x-1)(x-2) + (x%2B1)(x%2B2) – (x%2B2)(x-2)\). Expand: \(x^2-3x%2B2 + x^2%2B3x%2B2 – (x^2-4) = x^2-3x%2B2 + x^2%2B3x%2B2 – x^2 %2B 4 = x^2 %2B 8\). Equation \(\dfrac{x^2%2B8}{(x%2B2)(x-2)}=0\Rightarrow x^2%2B8=0\Rightarrow\) no real solutions (domain excludes \(x=\pm2\)).

No real solution.

Practice Questions (20) — With Difficulty & Collapsible Solutions

1. Medium Solve \(x^2-7x%2B10=0\).
   Solution

   Factor: \((x-5)(x-2)=0\Rightarrow x=2,5\).

   \(x=2,5\)

2. Ivy If \(q(x)=x^2-4x%2Bk\) has roots differing by 6, find \(k\).
   Solution

   Let roots \(r,s\) with \(r-s=6\) and sum \(r%2Bs=4\). Then \(r=2%2B3,\ s=2-3\Rightarrow rs=(2%2B3)(2-3)=-5\). So \(k=-rs=5\).

   \(k=5\)

3. MIT Minimum of \(y=2x^2-12x%2B29\) and the \(x\) where it occurs.
   Solution

   Vertex at \(x=\tfrac{12}{2\cdot2}=3\). Min value \(y(3)=18-36%2B29=11\).

   Minimum 11 at \(x=3\).

   
4. Ivy Solve \(x^2-5x=14\) by completing the square.
   Solution

   \(x^2-5x%2B(\tfrac{5}{2})^2=14%2B(\tfrac{5}{2})^2\Rightarrow (x-\tfrac{5}{2})^2=\tfrac{81}{4}\Rightarrow x=\tfrac{5\pm9}{2}=7,-2\).

   \(x=7,-2\)

5. Medium Vertex of \(y=-(x-3)^2%2B7\).
   Solution

   Form \(-(x-h)^2%2Bk\) ⇒ vertex \((h,k)=(3,7)\).

   Vertex \((3,7)\)

6. Ivy A quantity grows \(9\%\) per month. By what factor after 12 months?
   Solution

   Factor \(1.09^{12}\approx2.813\).

   \(\approx 2.813\times\)

   
7. Medium Solve \(\sqrt{2x%2B3}=x\).
   Solution

   Need \(x\ge0\). Square: \(2x%2B3=x^2\Rightarrow x^2-2x-3=0\Rightarrow (x-3)(x%2B1)=0\Rightarrow x=3,-1\). Check domain ⇒ \(x=3\).

   \(x=3\)

8. MIT Halving time for model \(N(t)=N_0(0.84)^t\).
   Solution

   Set \(0.5=(0.84)^t\Rightarrow t=\dfrac{\ln0.5}{\ln0.84}\approx4.17\).

   \(\approx 4.17\) time units

   
9. Ivy For \(y=500(1.08)^t\), find \(t\) when \(y=740\).
   Solution

   \(500(1.08)^t=740\Rightarrow (1.08)^t=1.48\Rightarrow t=\dfrac{\ln1.48}{\ln1.08}\approx4.66\).

   \(t\approx4.66\)

   
10. Medium Domain of \(y=\sqrt{5-2x}\).
    Solution

    Require \(5-2x\ge0\Rightarrow x\le2.5\).

    Domain: \(( -\infty,\ 2.5 ]\)

11. Medium Simplify \(\dfrac{x^2-1}{x^2-2x-3}\) and state restrictions.
    Solution

    \(x^2-1=(x-1)(x%2B1)\), \(x^2-2x-3=(x-3)(x%2B1)\). Simplifies to \(\dfrac{x-1}{x-3}\), restrictions \(x\ne -1,3\).

    \(\dfrac{x-1}{x-3}\); \(x\ne -1,3\)

    
12. Ivy Solve \(\dfrac{3}{x-1}-\dfrac{2}{x%2B1}=1\).
    Solution

    LCD \((x-1)(x%2B1)\). Get \(3(x%2B1)-2(x-1)=(x-1)(x%2B1)\Rightarrow x%2B5=x^2-1\Rightarrow x^2-x-6=0\Rightarrow x=3,-2\) (exclude \(\pm1\)).

    \(x=3,-2\)

13. MIT If \(\dfrac{ax%2B4}{x-2}\equiv 3+\dfrac{10}{x-2}\), find \(a\).
    Solution

    Rewrite RHS: \(\dfrac{3(x-2)%2B10}{x-2}=\dfrac{3x-6%2B10}{x-2}=\dfrac{3x%2B4}{x-2}\). Match numerators ⇒ \(a=3\).

    \(a=3\)

14. Ivy For \(f(x)=x^3-2x\), determine end behavior and real zeros.
    Solution

    Zeros at \(x=0,\pm\sqrt2\). As \(x\to\infty\), \(f\to\infty\); as \(x\to-\infty\), \(f\to-\infty\) (odd-degree, positive leading).

    Zeros: \(0,\pm\sqrt2\); end behavior up/right, down/left.

15. Medium Find horizontal asymptote of \(y=\dfrac{2x^2%2B1}{x^2-5}\).
    Solution

    Same degree: ratio of leading coefficients ⇒ \(y=2\).

    Horizontal asymptote: \(y=2\)

    Inverses & Composition