- The statement "The graph is narrower than the graph of \(f(x) = x^2\)" is true because the lead coefficient \(|a| = 6 > 1\).
- The statement "The graph is the same as the graph of \(f(x) = 6x^2 + 12x - 3\)" is true because expanding \(6(x+1)^2 - 9\) yields \(6x^2 + 12x - 3\).
- The statement "The graph opens upward" is true because the lead coefficient \(a = 6\) is positive.
- The three true statements are:
- The graph is narrower than the graph of \(f(x) = x^2\).
- The graph is the same as the graph of \(f(x) = 6x^2 + 12x - 3\).
- The graph opens upward.
Explanation
- Analysis of the function \(f(x) = 6(x+1)^2 - 9\)
The function \(f(x) = 6(x+1)^2 - 9\) is given in vertex form \(f(x) = a(x-h)^2 + k\), where \(a = 6\), \(h = -1\), and \(k = -9\). We will evaluate each statement to determine which three are correct.
- Evaluating the Statements
- Statement 1: The graph is narrower than the graph of \(f(x) = x^2\).
The width of a parabola \(y = ax^2\) depends on the absolute value of the lead coefficient \(a\). If \(|a| > 1\), the parabola is narrower than \(y = x^2\). Here, \(|a| = |6| = 6\). Since \(6 > 1\), the graph is indeed narrower than \(x^2\). This statement is true.
- Statement 2: The graph is the same as the graph of \(f(x) = 6x^2 + 12x - 3\).
We expand the given vertex form:
\[f(x) = 6(x+1)^2 - 9\]
\[f(x) = 6(x^2 + 2x + 1) - 9\]
\[f(x) = 6x^2 + 12x + 6 - 9\]
\[f(x) = 6x^2 + 12x - 3\]
The expanded form matches the expression in the statement. This statement is true.
- Statement 3: The graph is obtained by shifting the graph of \(f(x) = 6(x+1)^2\) up \(9\) units.
The function \(f(x) = 6(x+1)^2 - 9\) is obtained by taking the graph of \(g(x) = 6(x+1)^2\) and shifting it down by \(9\) units because of the \(-9\). Therefore, the statement saying it is shifted "up 9 units" is false.
- Statement 4: The graph opens upward.
The direction of opening is determined by the sign of the lead coefficient \(a\). Since \(a = 6 > 0\), the parabola opens upward. This statement is true.
- Statement 5: The vertex is \((1, -9)\).
In the vertex form \(f(x) = a(x-h)^2 + k\), the vertex is \((h, k)\).
Comparing \(f(x) = 6(x+1)^2 - 9\) to \(a(x-h)^2 + k\), we have \(h = -1\) and \(k = -9\). The vertex is \((-1, -9)\). The statement claims the vertex is \((1, -9)\), which is false.
- Conclusion
Based on our evaluation:
- The graph is narrower than \(f(x) = x^2\) (True)
- The graph is the same as \(f(x) = 6x^2 + 12x - 3\) (True)
- The graph opens upward (True)
The other two statements are false. The three correct statements are the first, second, and fourth.
Examples
Graphs are essential in engineering and architecture for modeling the strength and shape of structures. For example, when designing an arch for a bridge, architects use quadratic equations to determine its curvature. A parabola that is "narrower" (a larger absolute value of \(a\)) might represent a steeper arch, while the vertex provides the highest or lowest point of the structure, which is crucial for determining clearance and stress distribution. Knowing these properties helps engineers ensure the stability and safety of their designs.