At Westonci.ca, we provide reliable answers to your questions from a community of experts. Start exploring today! Explore thousands of questions and answers from a knowledgeable community of experts on our user-friendly platform. Join our Q&A platform to connect with experts dedicated to providing accurate answers to your questions in various fields.
Sagot :
To find the value of \( a \) in the equation of the quadratic function, follow these steps:
1. Identify the form of the quadratic function:
The zeros of the quadratic function are given as \( -8 \) and \( 4 \). We can start with the form:
[tex]\[ f(x) = a(x + 8)(x - 4) \][/tex]
where \( a \) is a constant that we need to determine.
2. Determine the vertex:
A vertex is a key feature of a parabola formed by a quadratic function. Given that the maximum point is \( (-2, 18) \), we know the vertex \( (-2, 18) \) lies on the graph of the quadratic function.
3. Substitute the vertex coordinates:
Substitute \( x = -2 \) and \( f(x) = 18 \) into the function to solve for \( a \).
[tex]\[ f(-2) = a((-2) + 8)((-2) - 4) = 18 \][/tex]
4. Simplify the equation:
Calculate the values inside the parentheses first:
[tex]\[ f(-2) = a(6)(-6) = 18 \][/tex]
Simplifying further,
[tex]\[ a \cdot 6 \cdot (-6) = 18 \][/tex]
[tex]\[ -36a = 18 \][/tex]
5. Solve for \( a \):
Isolate \( a \) by dividing both sides of the equation by \(-36\):
[tex]\[ a = \frac{18}{-36} = -\frac{1}{2} \][/tex]
Hence, the value of \( a \) in the quadratic function's equation is:
[tex]\[ \boxed{-\frac{1}{2}} \][/tex]
1. Identify the form of the quadratic function:
The zeros of the quadratic function are given as \( -8 \) and \( 4 \). We can start with the form:
[tex]\[ f(x) = a(x + 8)(x - 4) \][/tex]
where \( a \) is a constant that we need to determine.
2. Determine the vertex:
A vertex is a key feature of a parabola formed by a quadratic function. Given that the maximum point is \( (-2, 18) \), we know the vertex \( (-2, 18) \) lies on the graph of the quadratic function.
3. Substitute the vertex coordinates:
Substitute \( x = -2 \) and \( f(x) = 18 \) into the function to solve for \( a \).
[tex]\[ f(-2) = a((-2) + 8)((-2) - 4) = 18 \][/tex]
4. Simplify the equation:
Calculate the values inside the parentheses first:
[tex]\[ f(-2) = a(6)(-6) = 18 \][/tex]
Simplifying further,
[tex]\[ a \cdot 6 \cdot (-6) = 18 \][/tex]
[tex]\[ -36a = 18 \][/tex]
5. Solve for \( a \):
Isolate \( a \) by dividing both sides of the equation by \(-36\):
[tex]\[ a = \frac{18}{-36} = -\frac{1}{2} \][/tex]
Hence, the value of \( a \) in the quadratic function's equation is:
[tex]\[ \boxed{-\frac{1}{2}} \][/tex]
Visit us again for up-to-date and reliable answers. We're always ready to assist you with your informational needs. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Get the answers you need at Westonci.ca. Stay informed with our latest expert advice.