Question
The parabola has a vertex at (3, 2), a x-intercept at (5, 0), and a y-intercept at (0, -2).
Us the graph to write the equation of the quadratic function in vertex form.
y = -1/2(x - _)^2 + _
Us the graph to write the equation of the quadratic function in vertex form.
y = -1/2(x - _)^2 + _
Answers
GPT-4o mini
To find the equation of the quadratic function in vertex form, which is given by the equation:
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, we can follow these steps:
1. Identify the vertex: From the information given, the vertex is at \((3, 2)\). Therefore, \(h = 3\) and \(k = 2\).
2. Substitute the vertex into the equation:
\[ y = a(x - 3)^2 + 2 \]
3. To find the value of \(a\), we can use one of the known points. Since we have an x-intercept at \((5, 0)\), we can substitute \(x = 5\) and \(y = 0\) into the equation:
\[ 0 = a(5 - 3)^2 + 2 \]
4. Simplifying this:
\[ 0 = a(2)^2 + 2 \]
\[ 0 = 4a + 2 \]
\[ 4a = -2 \]
\[ a = -\frac{2}{4} = -\frac{1}{2} \]
5. Now we have \(a = -\frac{1}{2}\), so we can write the equation as:
\[ y = -\frac{1}{2}(x - 3)^2 + 2 \]
Thus, in the form \(y = -\frac{1}{2}(x - _)^2 + _\), we have:
- The first blank (h) is **3**.
- The second blank (k) is **2**.
Putting this all together, the final equation of the quadratic function in vertex form is:
\[ y = -\frac{1}{2}(x - 3)^2 + 2 \]
\[ y = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, we can follow these steps:
1. Identify the vertex: From the information given, the vertex is at \((3, 2)\). Therefore, \(h = 3\) and \(k = 2\).
2. Substitute the vertex into the equation:
\[ y = a(x - 3)^2 + 2 \]
3. To find the value of \(a\), we can use one of the known points. Since we have an x-intercept at \((5, 0)\), we can substitute \(x = 5\) and \(y = 0\) into the equation:
\[ 0 = a(5 - 3)^2 + 2 \]
4. Simplifying this:
\[ 0 = a(2)^2 + 2 \]
\[ 0 = 4a + 2 \]
\[ 4a = -2 \]
\[ a = -\frac{2}{4} = -\frac{1}{2} \]
5. Now we have \(a = -\frac{1}{2}\), so we can write the equation as:
\[ y = -\frac{1}{2}(x - 3)^2 + 2 \]
Thus, in the form \(y = -\frac{1}{2}(x - _)^2 + _\), we have:
- The first blank (h) is **3**.
- The second blank (k) is **2**.
Putting this all together, the final equation of the quadratic function in vertex form is:
\[ y = -\frac{1}{2}(x - 3)^2 + 2 \]