High School Math 3

Chapter at a Glance

• Chapter: 1
• Strand: Pure — Change
• Difficulty: Level 4 of 5
• Builds on: The quadratic formula and the discriminant, Transforming functions, Polynomials

Try This First

Try this short activity before reading the lesson.

> Take $y=(x+1)(x-2)^2$. Before reading anything, plug in $x=-2$, $x=0$, $x=3$ and just write the sign of each answer. Then mark where $y=0$. You now have enough to draw the shape; do it, and keep the sketch.

Math lab: Predict the shape from signs

Predict first. For $f(x) = (x - 1)(x + 2)$, predict whether the graph is above or below the axis between its zeros.

<div class="math-model"><p><strong>Sign chart. Zeros at −2 and 1.</strong></p><div class="model-bar"><span class="model-cell">x &lt; −2: +</span><span class="model-cell">−2 &lt; x &lt; 1: −</span><span class="model-cell">x &gt; 1: +</span></div><p class="model-caption">Intervals are schematic, not drawn to scale.</p></div>

Model and solve. Test $x = -3, 0, 2$: outputs are $4, -2, 4$. The product is negative between the zeros. Plot the intercepts $(-2,0)$, $(1,0)$ and $(0,-2)$, then sketch the upward-opening parabola.

Spot the bug / explain. A learner says two factors always make a positive product. Use $x=0$ as a counterexample.

Your turn. Find zeros and the negative interval for $(x-2)(x+1)$.

Check after trying. Zeros: $-1$ and $2$; negative for $-1 < x < 2$.

The Big Idea

Far from the origin, only the leading term matters: at $x=1000$, the $x^4$ in $3x^4-5x^3+x$ is a thousand times the next term. So the degree and the sign of the leading coefficient decide the two ends. Even degree: both ends go the same way. Odd degree: opposite ways. Negative leading coefficient flips both.

Near the axis, the factors matter. A zero from a factor to an odd power crosses the axis; a zero from an even power touches and turns back, a bounce, because the sign cannot change there. Between zeros the sign is fixed, so one test point per interval settles it.

The graph can turn at most $n-1$ times for degree $n$. That is a ceiling, not a count.

The leading term decides the ends and the factors decide the middle.

Words to Know

• End behavior: Where the graph heads as $x\to\infty$ and as $x\to-\infty$. Decided by the leading term alone.
• Multiplicity: The power on a factor. $(x-1)^2$ gives the zero $1$ multiplicity $2$. Odd: cross. Even: bounce.
• Turning point: A place where the graph changes from rising to falling or back. Degree $n$ allows at most $n-1$.

Formulas and Theorems

• End behavior: $p(x)\sim a_n x^n \text{ for large } |x|$
• Note: Sign of $a_n$ and parity of $n$ decide where the ends point.
• Multiplicity: $(x-r)^m$
• Note: Crosses at $r$ if $m$ is odd, bounces if $m$ is even.
• Turning points: $\le n-1$
• Note: A degree-$n$ polynomial turns at most $n-1$ times.

Outcomes and Milestones

• Students can: read degree, leading coefficient and end behavior from standard or factored form; list zeros with multiplicity and say cross or bounce at each; sketch from factored form with one sign check per interval; state the maximum number of turning points as $n-1$.
• Watch for: $(x+3)^2$ read as a zero at $x=3$; “degree 4 so four turns”; a bounce drawn as a crossing.
• Ready when: $p(x)=-(x+2)(x-1)^2(x-4)$ sketched with the correct ends, a bounce at $1$, and the sign of $p(0)$ stated without a calculator.

Worked Examples and Your Turn

Study each example, then try the paired problem.

Example 1: The ends from the leading term

Problem: Describe the end behavior of $p(x)=3x^4-5x^3+x$.

Solution:

• Find the leading term: $3x^4$. Why: for large $|x|$ it swamps everything else, so it alone sets the ends.
• Degree $4$ is even, coefficient $3$ is positive. Why: $x^4$ is positive on both sides, and multiplying by $3$ keeps it positive.

Answer: Both ends go up: $p(x)\to\infty$ as $x\to\infty$ and as $x\to-\infty$.

> Verification & Check: $p(10)=30000-5000+10>0$ and $p(-10)=30000+5000-10>0$.

Your Turn:

• Prompt: End behavior of $q(x)=-2x^3+7x$?
• Answer: Odd degree, negative coefficient: left end up, right end down.

Example 2: Sketch from factored form

Problem: Sketch $p(x)=-(x+2)(x-1)^2(x-4)$ and state the sign of $p(0)$ without a calculator.

The chapter continues in the book.