Pure Mathematics Practice: Graphing Techniques — Answers

Questions: Graphing Techniques Drills.

Question 1

(a) One valid order is:

1. translate the graph 3 units to the right;
2. stretch it vertically by scale factor 2;
3. translate it 1 unit down.

(b) The input coordinate changes from $1$ to $1+3=4$, while the output changes from $4$ to $2(4)-1=7$. The corresponding point is

$$(4,7).$$

(c) Since multiplication by the positive number 2 preserves endpoint order,

$$2[-2,5]-1=[-5,9].$$

Question 2

(a)

$$\frac{3x-1}{x+2}=3-\frac{7}{x+2}.$$

(b) The vertical asymptote is $x=-2$ and the horizontal asymptote is $y=3$.

(c) The $x$-intercept is $(1/3,0)$. Setting $x=0$ gives the $y$-intercept $(0,-1/2)$.

(d)

Because the translated reciprocal coefficient is negative, the branch for $x>-2$ lies below $y=3$, while the branch for $x<-2$ lies above $y=3$.

Question 3

(a)

$$g(x)=(x-1)(x-3)=(x-2{)}^2-1.$$

The roots are $1$ and $3$, and the vertex is $(2,-1)$.

(b) Reflect the portion of $y=g(x)$ below the $x$-axis in the $x$-axis. The reflected graph has sharp, non-differentiable points at $(1,0)$ and $(3,0)$.

(c) The equation $|g(x)|=3$ means $g(x)=3$ or $g(x)=-3$.

For $g(x)=3$,

$$x^2-4x=0,$$

so $x=0$ or $x=4$. For $g(x)=-3$,

$$x^2-4x+6=0,$$

whose discriminant is negative. Therefore

$$x=0\text{or}x=4.$$

Question 4

(a) The axis of symmetry is

$$x=2.$$

(b) The maximum is $5$ at $x=2$. At both endpoints,

$$h(-1)=h(5)=-4.$$

Hence

$$R_h=[-4,5].$$

(c)

$$5-(x-2{)}^2\ge 1⟺(x-2{)}^2\le 4.$$

Thus

$$-2\le x-2\le 2,$$

so, within the stated domain,

$$0\le x\le 4.$$