Q. 4 - Precalculus Enhanced with Graphing Utilities

Question

In Problems 4 and 5,

(a) Determine whether the graph opens up or down.

(b) Determine the vertex of the graph of the quadratic function.

(c) Determine the axis of symmetry of the graph of the quadratic

function.

(d) Determine the intercepts of the graph of the quadratic function.

(e) Use the information in parts (a)–(d) to graph the quadratic

function.

(f) Based on the graph, determine the domain and the range of

the quadratic function.

(g) Based on the graph, determine where the function is

increasing and where it is decreasing.

$f\left(x\right)=3x^2-12x+4$

Step-by-Step Solution

Answer

(a) Graph is open up.

(b) vertex$=(2,8)$

(c) Axis of symmetry $x=2$

(d) y-intercept $(0,4)$, x-intercept $(\frac{6\pm 2\sqrt{6}}{3},0)$

(e) 

(f) Domain$=(-\infty ,\infty )$ and Range$=[-8,\infty )$

(g) Function is decreasing in$(-\infty ,2]$ and increasing in $[2,\infty )$

Part (a) Step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (a) Step 2. Calculation

Comparing the given function with $f\left(x\right)=a{x}^2+bx+c$ we get,

$a=3,b=-12 \& c=4$

Hence $a>0$ then the graph of the given function is opened up.

Part (b) Step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (b) Step 2. Calcuation

Since $f\left(x\right)=3x^2-12x+4$

Then the vertex is at $(2,8)$

Part (c) Step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (c) Step 2 Calculation

Since vertex is at $(2,8)$ 

Then the axis of symmetry is $x=2$

Part (d) Step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (d) Step 2. Calculation

y-intercept, $f\left(0\right)=3.0-12.0+4$

$=4$

Then y-intercept is  $(0,4)$

For x-intercept ${12}^2-4.3.4>0$

Then it has two x-intercepts.

Now, $3x^2-12x+4=0$

Therefore $x=\frac{6\pm 2\sqrt{6}}{3}$

Then x-intercept is $(\frac{6\pm 2\sqrt{6}}{3},0)$

Part (e) Step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (e) Step 2. Graph

Part (f) Step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (f) Step 2. Calculation

The domain of the given function is $(-\infty ,\infty )$ and Range is $[-8,\infty )$.

Part (g) step 1. Given Information

$f\left(x\right)=3x^2-12x+4$

Part (f) Step 2. Calculation

By the graph, the function is decreasing in the interval $(-\infty ,2]$ and the function is increasing in the interval $[2,\infty )$.