Question

Find the points of intersection of the graphs of the finctinns \(f(r)=r^{2}+3 r+7\) and \(p(r)=-2 r+3\)

Short answer

The points of intersection are ( -1, 5) and (3, -3).

Step-by-step solution

- Set the functions equal to each other

To find the points of intersection, set the functions equal to each other:

- Rearrange the equation

Rearrange the equation to form a standard quadratic equation:

- Solve the quadratic equation

Use the quadratic formula to find the values of r: Use the quadratic formula: where which is simplified to: Substituting into the quadratic formula, we get: Therefore, we have: Simplifying, we get: Therefore: Thus, the solutions for r are:

- Solve for the intersection points

Substitute the values of r into the linear equation p(r) = -2r +3 to find the corresponding y-values: For r = -1: For r = 3: Thus, the points of intersection are: and

Key concepts

Quadratic Equations

A quadratic equation is an equation of the form \( ax^2 + bx + c = 0 \), where \( a \), \( b \), and \( c \) are constants, and \( x \) is the variable. The highest power of \( x \) is 2, which makes it a quadratic equation.

Quadratic equations can be solved by various methods, such as:

• Factoring,
• Completing the square,
• Using the quadratic formula.

The solutions to a quadratic equation are where the graph of the quadratic function intersects the x-axis. These points are called the roots of the equation.

In this exercise, the quadratic equation is formed by setting the quadratic function \( f(r) = r^2 + 3r + 7 \) equal to the linear function \( p(r) = -2r + 3 \).

Linear Equations

A linear equation is an equation of the form \( y = mx + b \), where \( m \) is the slope, and \( b \) is the y-intercept. The graph of a linear equation is a straight line.

Linear equations can be easily solved by isolating the variable. In this exercise, the linear equation is \( p(r) = -2r + 3 \).

To find the points of intersection, we need to set the linear equation equal to the quadratic equation and solve for the variable \( r \). This process will reveal where the two graphs intersect.

Quadratic Formula

The quadratic formula is a powerful tool for solving quadratic equations. It is given by:
\( r = \frac{ -b \pm \sqrt{ b^2 - 4ac } }{ 2a } \)

Here, \( a \), \( b \), and \( c \) are the coefficients of the quadratic equation \( ax^2 + bx + c = 0 \).

The term under the square root, \( b^2 - 4ac \), is called the discriminant. It determines the nature of the roots:

• If \( b^2 - 4ac > 0 \), the equation has two distinct real roots.
• If \( b^2 - 4ac = 0 \), the equation has one real root (a repeated root).
• If \( b^2 - 4ac < 0 \), the equation has no real roots (the roots are complex numbers).

In this exercise, after setting \( f(r) = r^2 + 3r + 7 \) equal to \( p(r) = -2r + 3 \), we rearrange it to form a standard quadratic equation and use the quadratic formula to find the values of \( r \).

Points of Intersection

The points of intersection of two functions are the values of \( x \) (or \( r \) in this case) where the two graphs meet, and the corresponding y-values.

To find these points, we follow these steps:

• Set the two functions equal to each other.
• Rearrange the equation to form a standard quadratic equation.
• Solve the quadratic equation using a suitable method, such as the quadratic formula.
• Substitute the values of the variable back into one of the original functions to find the corresponding y-values.

In this exercise:

1. We set \( r^2 + 3r + 7 = -2r + 3 \).
2. Rearrange to form \( r^2 + 5r + 4 = 0 \).
3. Use the quadratic formula to find the values of \( r \), which are \( r = -1 \) and \( r = -4 \).
4. Substitute these values into the linear equation \( p(r) = -2r + 3 \) to find the y-values.

The points of intersection are: \( r = -1, y = 5 \) and \( r = 3, y = -3 \).