Question

1. $y = (x + 5)(x + 7)$ (and a coordinate grid)

Explanation:

To analyze the graph of \( y=(x + 5)(x + 7) \), we can follow these steps:

Step 1: Find the x - intercepts

The x - intercepts occur when \( y = 0 \). So we set \( (x + 5)(x + 7)=0 \)
Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \)
If \( x+5=0 \), then \( x=-5 \)
If \( x + 7=0 \), then \( x=-7 \)
So the x - intercepts are at the points \( (-7,0) \) and \( (-5,0) \)

Step 2: Find the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation \( y=(x + 5)(x + 7) \)
\( y=(0 + 5)(0+7)=5\times7 = 35 \)
So the y - intercept is at the point \( (0,35) \)

Step 3: Expand the equation to find the vertex form (optional, but helps in graphing)

First, expand \( y=(x + 5)(x + 7) \) using the FOIL method.
\( y=x^{2}+7x+5x + 35=x^{2}+12x + 35 \)
The vertex of a parabola in the form \( y=ax^{2}+bx + c \) has its x - coordinate given by \( x=-\frac{b}{2a} \)
For \( y=x^{2}+12x + 35 \), \( a = 1 \), \( b = 12 \)
\( x=-\frac{12}{2\times1}=-6 \)
Substitute \( x=-6 \) into the equation \( y=x^{2}+12x + 35 \)
\( y=(-6)^{2}+12\times(-6)+35=36-72 + 35=-1 \)
So the vertex of the parabola is at the point \( (-6,-1) \)

Step 4: Determine the direction of the parabola

Since the coefficient of \( x^{2} \) in the expanded form \( y=x^{2}+12x + 35 \) is \( a = 1>0 \), the parabola opens upwards.

To graph the parabola:

• Plot the x - intercepts \( (-7,0) \) and \( (-5,0) \)
• Plot the y - intercept \( (0,35) \)
• Plot the vertex \( (-6,-1) \)
• Then, draw a smooth parabola passing through these points, opening upwards.

Answer:

To analyze the graph of \( y=(x + 5)(x + 7) \), we can follow these steps:

Step 1: Find the x - intercepts

The x - intercepts occur when \( y = 0 \). So we set \( (x + 5)(x + 7)=0 \)
Using the zero - product property, if \( ab = 0 \), then either \( a = 0 \) or \( b = 0 \)
If \( x+5=0 \), then \( x=-5 \)
If \( x + 7=0 \), then \( x=-7 \)
So the x - intercepts are at the points \( (-7,0) \) and \( (-5,0) \)

Step 2: Find the y - intercept

The y - intercept occurs when \( x = 0 \). Substitute \( x = 0 \) into the equation \( y=(x + 5)(x + 7) \)
\( y=(0 + 5)(0+7)=5\times7 = 35 \)
So the y - intercept is at the point \( (0,35) \)

Step 3: Expand the equation to find the vertex form (optional, but helps in graphing)

First, expand \( y=(x + 5)(x + 7) \) using the FOIL method.
\( y=x^{2}+7x+5x + 35=x^{2}+12x + 35 \)
The vertex of a parabola in the form \( y=ax^{2}+bx + c \) has its x - coordinate given by \( x=-\frac{b}{2a} \)
For \( y=x^{2}+12x + 35 \), \( a = 1 \), \( b = 12 \)
\( x=-\frac{12}{2\times1}=-6 \)
Substitute \( x=-6 \) into the equation \( y=x^{2}+12x + 35 \)
\( y=(-6)^{2}+12\times(-6)+35=36-72 + 35=-1 \)
So the vertex of the parabola is at the point \( (-6,-1) \)

Step 4: Determine the direction of the parabola

Since the coefficient of \( x^{2} \) in the expanded form \( y=x^{2}+12x + 35 \) is \( a = 1>0 \), the parabola opens upwards.

To graph the parabola:

• Plot the x - intercepts \( (-7,0) \) and \( (-5,0) \)
• Plot the y - intercept \( (0,35) \)
• Plot the vertex \( (-6,-1) \)
• Then, draw a smooth parabola passing through these points, opening upwards.