Questions: y=x^2-8, y=x+4 smaller x-value (x, y)= larger x-value (x, y)=

Transcript text: \[ y=x^{2}-8, \quad y=x+4 \] smaller $x$-value $(x, y)=$ $\square$ ) larger $x$-value $(x, y)=$ $\square$ )

Solution

Solution Steps

To find the points of intersection between the two equations $ y = x^2 - 8 $ and $ y = x + 4 $, we need to set the equations equal to each other and solve for $ x $. This will give us the $ x $-values where the curves intersect. Then, we can substitute these $ x $-values back into either equation to find the corresponding $ y $-values.

Solution Approach

Set the equations equal to each other: $ x^2 - 8 = x + 4 $.
Rearrange the equation to form a standard quadratic equation: $ x^2 - x - 12 = 0 $.
Solve the quadratic equation for $ x $ to find the intersection points.
Substitute the $ x $-values back into one of the original equations to find the corresponding $ y $-values.
Identify the smaller and larger $ x $-values and their corresponding $ y $-values.

Step 1: Set the Equations Equal to Each Other

To find the points of intersection, we set the equations equal to each other: \[ x^2 - 8 = x + 4 \]

Step 2: Rearrange to Form a Quadratic Equation

Rearrange the equation to form a standard quadratic equation: \[ x^2 - x - 12 = 0 \]

Step 3: Solve the Quadratic Equation

Solve the quadratic equation for $ x $: \[ x = -3 \quad \text{and} \quad x = 4 \]

Step 4: Find Corresponding $ y $-Values

Substitute the $ x $-values back into one of the original equations, $ y = x + 4 $, to find the corresponding $ y $-values: \[ \text{For } x = -3, \quad y = -3 + 4 = 1 \] \[ \text{For } x = 4, \quad y = 4 + 4 = 8 \]

Step 5: Identify the Points of Intersection

The points of intersection are: \[ (-3, 1) \quad \text{and} \quad (4, 8) \]