91Ó°ÊÓ

A quadratic function is represented by the formula \( f(x) = ax^2 + bx + c \). These functions graph as parabolas that open either upwards or downwards. The sign of the coefficient \( a \) determines the direction of the parabola:

• If \( a > 0 \), the parabola opens upwards.
• If \( a < 0 \), the parabola opens downwards.

In the given exercise, we have two quadratic functions: \( f(x) = x^2 - 4x + 10 \) and \( g(x) = 2x^2 - 12x + 14 \). These functions create two distinct parabolas when plotted. Understanding the characteristics of quadratic functions helps predict their shape and behavior on a graph.

Intersection points are crucial for determining where two functions meet on a graph. To find these points, set the equations equal to each other: \( f(x) = g(x) \). For our functions, \( x^2 - 4x + 10 = 2x^2 - 12x + 14 \). Rearrange this to \( x^2 - 8x + 4 = 0 \).

Quadratic Formula

Use the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \) to solve for \( x \). Here, \( a = 1 \), \( b = -8 \), and \( c = 4 \). Calculating gives \( x = 6 \pm 2\sqrt{2} \). Hence, the functions intersect approximately at \( x \approx 3.17 \) and \( x \approx 8.83 \). These points help us understand the interval within which we calculate the area between the curves.

The integral helps calculate the area between two curves. Once you identify where the curves intersect, you use these points to set the bounds of the integral. For our problem, despite the intersection, the exercise restricts the calculation between \( x = 1 \) and \( x = 7 \).

Setting Up the Integral

To find the area between \( f(x) \) and \( g(x) \), compute the integral of the difference \( |f(x) - g(x)| \). For the interval \( 1 \) to \( 7 \), \( f(x) - g(x) = -(x^2 - 8x + 4) \). Simplify to get \( x^2 - 8x + 4 \), since this is positive over our interval.Use the calculated integral \( \int_{1}^{7} (x^2 - 8x + 4) \ dx \). For each term inside the integral, apply the power rule: \( \int x^n \ dx = \frac{x^{n+1}}{n+1} \). Thus, \( \frac{x^3}{3} - 4x^2 + 4x \). Evaluate it from 1 to 7 to get the area.

Graph sketching involves plotting both functions on the same axes to visualize their interaction. This step is essential for identifying regions of interest and gaining intuitive insight into the problem. Start by choosing several points for each function by substituting values into the equations.

Steps for Graph Sketching

• Calculate a few values for \( f(x) \) and \( g(x) \) by substituting \( x \) values, especially near suspected intersections or turning points.
• Plot these points carefully on the graph. The more points you calculate, the more accurate your sketch becomes.
• Draw smooth curves through these points since both functions are quadratic and exhibit a parabolic shape.
• Identify and shade the region between the curves from \( x = 1 \) to \( x = 7 \) as specified.

Graph sketching aids in visualizing the complexity of the area being calculated and solidifies understanding of the functions' interactions.