Part of the graph of the function $f(x)=(x-1)(x+7)$ is shown below.
Which statements about the function are true? Select three options.
A. The vertex of the function is at $(-4,-15)$.
B. The vertex of the function is at ( $-3,-16$ ).
C. The graph is increasing on the interval $x>-3$.
D. The graph is positive only on the intervals where $x<$ -7 and where $x>1$.
E. The graph is negative on the interval $x<-4$.
Answer (1)
Expand the function to get f ( x ) = x 2 + 6 x − 7 .
Calculate the vertex using x v = − 2 a b and y v = f ( x v ) , which gives the vertex as ( − 3 , − 16 ) .
Determine that the function is increasing for -3"> x > − 3 .
Find the roots of the function, x = − 7 and x = 1 , and determine that the function is positive for x < − 7 and 1"> x > 1 .
The three true statements are: vertex at ( − 3 , − 16 ) , increasing for -3"> x > − 3 , and positive for x < − 7 and 1"> x > 1 . -3; The graph is positive only on the intervals where x<-7 and where x>1}"> T h e v er t e x o f t h e f u n c t i o ni s a t ( − 3 , − 16 ) ; T h e g r a p hi s in cre a s in g o n t h e in t er v a l x > − 3 ; T h e g r a p hi s p os i t i v eo n l yo n t h e in t er v a l s w h ere x < − 7 an d w h ere x > 1
Explanation
Analyzing the Problem We are given the function f ( x ) = ( x − 1 ) ( x + 7 ) and its graph. We need to determine which of the given statements about the function are true. Let's analyze the function and the statements.
Expanding the Function First, let's expand the function: f ( x ) = ( x − 1 ) ( x + 7 ) = x 2 + 7 x − x − 7 = x 2 + 6 x − 7 . This is a quadratic function in the form f ( x ) = a x 2 + b x + c , where a = 1 , b = 6 , and c = − 7 .
Finding the Vertex The vertex of a parabola f ( x ) = a x 2 + b x + c is given by the coordinates ( x v , y v ) , where x v = − 2 a b and y v = f ( x v ) . In our case, x v = − 2 ( 1 ) 6 = − 3 . Now, let's find the y-coordinate of the vertex: y v = f ( − 3 ) = ( − 3 ) 2 + 6 ( − 3 ) − 7 = 9 − 18 − 7 = − 16 . Therefore, the vertex of the function is at ( − 3 , − 16 ) .
Determining the Increasing Interval The graph of the function is a parabola that opens upwards since 0"> a = 1 > 0 . The function is increasing to the right of the vertex, which means it is increasing on the interval -3"> x > − 3 .
Determining Positive and Negative Intervals The roots of the function are the values of x for which f ( x ) = 0 . We already have the factored form of the function: f ( x ) = ( x − 1 ) ( x + 7 ) . The roots are x = 1 and x = − 7 . Since the parabola opens upwards, the graph is positive (i.e., 0"> f ( x ) > 0 ) when x < − 7 or 1"> x > 1 , and negative (i.e., f ( x ) < 0 ) when − 7 < x < 1 .
Comparing with the Given Statements Now, let's compare our findings with the given statements:
The vertex of the function is at ( − 4 , − 15 ) . This is false, as we found the vertex to be at ( − 3 , − 16 ) .
The vertex of the function is at ( − 3 , − 16 ) . This is true.
The graph is increasing on the interval -3"> x > − 3 . This is true.
The graph is positive only on the intervals where x < − 7 and where 1"> x > 1 . This is true.
The graph is negative on the interval x < − 4 . This is false, as the graph is negative between -7 and 1.
Final Answer Therefore, the three true statements are:
The vertex of the function is at ( − 3 , − 16 ) .
The graph is increasing on the interval -3"> x > − 3 .
The graph is positive only on the intervals where x < − 7 and where 1"> x > 1 .
Examples
Understanding the properties of quadratic functions, such as finding the vertex and intervals where the function is positive or increasing, is crucial in various real-world applications. For example, engineers use this knowledge to design parabolic reflectors for satellite dishes or solar cookers, optimizing the focus of incoming signals or sunlight. Similarly, in business, understanding the vertex of a cost function can help determine the production level that minimizes costs and maximizes profits. These applications highlight the practical significance of analyzing quadratic functions.
