Exercise 49 Page 337

Practice makes perfect

a We want to graph the given quadratic function

f (x)

and label the

x -

and

y -

intercepts.

f (x) = 3 (x - 4)^{2} - 5

This function is given in the vertex form.

f (x) = a (x - k)^{2} + h

The coordinates of the vertex are

(k, h),

so the vertex of our graph is

(4, - 5) .

To sketch the graph accurately, we will make a table of values to help us find more points on the graph.

$x$	$3 (x - 4)^{2} - 5 = y$	$(x, y)$
$0$	$3 (0 - 4)^{2} - 5 = 43$	$(0, 43)$
$2$	$3 (2 - 4)^{2} - 5 = 7$	$(2, 7)$
$3$	$3 (3 - 4)^{2} - 5 = - 2$	$(3, - 2)$
$5$	$3 (5 - 4)^{2} - 5 = - 2$	$(5, - 2)$
$6$	$3 (6 - 4)^{2} - 5 = 7$	$(6, 7)$
$8$	$3 (8 - 4)^{2} - 5 = 43$	$(8, 43)$

Let's plot these points and the vertex in the coordinate plane. Then we can connect the points with a smooth curve.

Next, we need to label the

x -

and

y -

intercepts. Recall that the

x -

intercepts are the

x -

values where the graph crosses the

x -

axis, while the

y -

intercept is the

y -

value where the graph crosses the

y -

axis. We can find our intercepts by solving two equations.

f (x) = 0 f (0) = y

The first equation gives us the

x -

intercepts and the second one gives us the

y -

intercept. Let's start with the

x -

intercepts.

y = 0

Substitute

$y = 3 (x - 4)^{2} - 5$

3 (x - 4)^{2} - 5 = 0

AddEqn

$LHS + 5 = RHS + 5$

3 (x - 4)^{2} = 5

DivEqn

$LHS / 3 = RHS / 3$

(x - 4)^{2} = \frac{5}{3}

RadicalEqn

$2 LHS = 2 RHS$

(x - 4)^{2} = \frac{5}{3}

$a^{2} = \pm a$

\pm (x - 4) = \frac{5}{3}

Let's simplify the solutions! We can use a calculator to help calculate the exact values.

$(x - 4) = \frac{5}{3}$	$x = \frac{5}{3} + 4$	$x \approx 5.29$
$- (x - 4) = \frac{5}{3}$	$x = - \frac{5}{3} + 4$	$x \approx 2.71$

Let's label the $x -$ intercepts in our graph!

Our graph is coming along! Now let's find the

y -

intercept.

f (0) = y

Notice that we already found the value of

y

when

x = 0

in our table of values. We found that when

x = 0,

y = 43 .

Let's label it in our graph.

b We want to graph the given quadratic function and label the

x -

and

y -

intercepts.

f (x) = 2 x^{2} - 3 x - 5

In Part A, the function was given in vertex form, but this equation is given to us in standard form. Let's start by finding the coordinates of the vertex. To calculate the vertex, we need to think of