Question

1. graph each of the following - determine 5 or 3 points on the graph

a. $y = 3(x + 1)^2 - 23$
b. $y = x^2 - 10x + 26$

Explanation:

Part a: \( y = 3(x + 1)^2 - 23 \)

Step 1: Identify the vertex form

The equation \( y = 3(x + 1)^2 - 23 \) is in vertex form \( y = a(x - h)^2 + k \), where the vertex is \( (h, k) \). Here, \( h = -1 \) and \( k = -23 \), so the vertex is \( (-1, -23) \). This is our first point.

Step 2: Choose \( x \)-values around the vertex

Let's choose \( x = -2, 0, 1, -3 \) (values around \( x = -1 \)) to find corresponding \( y \)-values.

• For \( x = -2 \):

\( y = 3(-2 + 1)^2 - 23 = 3(-1)^2 - 23 = 3(1) - 23 = 3 - 23 = -20 \)
So the point is \( (-2, -20) \).

• For \( x = 0 \):

\( y = 3(0 + 1)^2 - 23 = 3(1)^2 - 23 = 3 - 23 = -20 \)
So the point is \( (0, -20) \).

• For \( x = 1 \):

\( y = 3(1 + 1)^2 - 23 = 3(2)^2 - 23 = 3(4) - 23 = 12 - 23 = -11 \)
So the point is \( (1, -11) \).

• For \( x = -3 \):

\( y = 3(-3 + 1)^2 - 23 = 3(-2)^2 - 23 = 3(4) - 23 = 12 - 23 = -11 \)
So the point is \( (-3, -11) \).

Now we have 5 points: \( (-3, -11) \), \( (-2, -20) \), \( (-1, -23) \), \( (0, -20) \), \( (1, -11) \).

Part b: \( y = x^2 - 10x + 26 \)

Step 1: Complete the square (or use vertex formula)

First, find the vertex. For a quadratic \( y = ax^2 + bx + c \), the \( x \)-coordinate of the vertex is \( x = -\frac{b}{2a} \). Here, \( a = 1 \), \( b = -10 \), so \( x = -\frac{-10}{2(1)} = 5 \).

Substitute \( x = 5 \) into the equation: \( y = (5)^2 - 10(5) + 26 = 25 - 50 + 26 = 1 \). So the vertex is \( (5, 1) \) (first point).

Step 2: Choose \( x \)-values around \( x = 5 \)

Let's choose \( x = 4, 6, 3, 7 \).

• For \( x = 4 \):

\( y = (4)^2 - 10(4) + 26 = 16 - 40 + 26 = 2 \)
Point: \( (4, 2) \).

• For \( x = 6 \):

\( y = (6)^2 - 10(6) + 26 = 36 - 60 + 26 = 2 \)
Point: \( (6, 2) \).

• For \( x = 3 \):

\( y = (3)^2 - 10(3) + 26 = 9 - 30 + 26 = 5 \)
Point: \( (3, 5) \).

• For \( x = 7 \):

\( y = (7)^2 - 10(7) + 26 = 49 - 70 + 26 = 5 \)
Point: \( (7, 5) \).

Now we have 5 points: \( (3, 5) \), \( (4, 2) \), \( (5, 1) \), \( (6, 2) \), \( (7, 5) \).

Answer:

Part a Points: \( (-3, -11) \), \( (-2, -20) \), \( (-1, -23) \), \( (0, -20) \), \( (1, -11) \)

Part b Points: \( (3, 5) \), \( (4, 2) \), \( (5, 1) \), \( (6, 2) \), \( (7, 5) \)