Question
Write the equation of the quadratic function in the graph in standard form. The axis of symmetry is at 3. -1.
The parabola has a y intercept at (0, 8) and a x intercept at (2, 0)
The parabola has a y intercept at (0, 8) and a x intercept at (2, 0)
Answers
GPT-4o mini
To write the equation of the quadratic function in standard form, we first note that the standard form of a quadratic function is given by:
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and width of the parabola.
From the information given:
- The axis of symmetry is at \(x = 3\), which means that the vertex has coordinates \((3, k)\) for some \(k\).
- The parabola has a \(y\)-intercept at \((0, 8)\).
- It also has an \(x\)-intercept at \((2, 0)\).
### Step 1: Find the value of \(k\)
We first need to determine the vertex. Since we have the axis of symmetry at \(x = 3\), let's evaluate the function at the \(y\)-intercept and \(x\)-intercept to find \(k\).
Using the x-intercept \((2, 0)\) to find the corresponding \(y\) value when \(x = 2\):
If we substitute \(x = 2\) into the vertex form \(f(2) = a(2 - 3)^2 + k = 0\) which simplifies to:
\[ 0 = a(1) + k \]
\[ k = -a \]
Next, using the y-intercept \((0, 8)\):
If we substitute \(x = 0\):
\[ f(0) = a(0 - 3)^2 + k = 8 \]
\[ 8 = a(9) + k \]
\[ 8 = 9a + k \]
### Step 2: Substitute \(k\) in terms of \(a\)
From \(k = -a\), substitute this into the equation from the y-intercept:
\[ 8 = 9a + (-a) \]
\[ 8 = 9a - a \]
\[ 8 = 8a \]
### Step 3: Solve for \(a\)
Solving for \(a\):
\[ a = 1 \]
### Step 4: Find \(k\)
Substituting \(a = 1\) back to find \(k\):
\[ k = -a = -1 \]
### Step 5: Write the vertex form of the equation
Now, we have \(a = 1\) and \((h, k) = (3, -1)\):
\[ f(x) = 1(x - 3)^2 - 1 \]
\[ f(x) = (x - 3)^2 - 1 \]
### Step 6: Convert to standard form
Now let's convert to standard form:
\[ f(x) = (x - 3)(x - 3) - 1 \]
\[ = (x^2 - 6x + 9) - 1 \]
\[ = x^2 - 6x + 8 \]
Thus, the quadratic function in standard form is:
\[ f(x) = x^2 - 6x + 8 \]
\[ f(x) = a(x - h)^2 + k \]
where \((h, k)\) is the vertex of the parabola, and \(a\) determines the direction and width of the parabola.
From the information given:
- The axis of symmetry is at \(x = 3\), which means that the vertex has coordinates \((3, k)\) for some \(k\).
- The parabola has a \(y\)-intercept at \((0, 8)\).
- It also has an \(x\)-intercept at \((2, 0)\).
### Step 1: Find the value of \(k\)
We first need to determine the vertex. Since we have the axis of symmetry at \(x = 3\), let's evaluate the function at the \(y\)-intercept and \(x\)-intercept to find \(k\).
Using the x-intercept \((2, 0)\) to find the corresponding \(y\) value when \(x = 2\):
If we substitute \(x = 2\) into the vertex form \(f(2) = a(2 - 3)^2 + k = 0\) which simplifies to:
\[ 0 = a(1) + k \]
\[ k = -a \]
Next, using the y-intercept \((0, 8)\):
If we substitute \(x = 0\):
\[ f(0) = a(0 - 3)^2 + k = 8 \]
\[ 8 = a(9) + k \]
\[ 8 = 9a + k \]
### Step 2: Substitute \(k\) in terms of \(a\)
From \(k = -a\), substitute this into the equation from the y-intercept:
\[ 8 = 9a + (-a) \]
\[ 8 = 9a - a \]
\[ 8 = 8a \]
### Step 3: Solve for \(a\)
Solving for \(a\):
\[ a = 1 \]
### Step 4: Find \(k\)
Substituting \(a = 1\) back to find \(k\):
\[ k = -a = -1 \]
### Step 5: Write the vertex form of the equation
Now, we have \(a = 1\) and \((h, k) = (3, -1)\):
\[ f(x) = 1(x - 3)^2 - 1 \]
\[ f(x) = (x - 3)^2 - 1 \]
### Step 6: Convert to standard form
Now let's convert to standard form:
\[ f(x) = (x - 3)(x - 3) - 1 \]
\[ = (x^2 - 6x + 9) - 1 \]
\[ = x^2 - 6x + 8 \]
Thus, the quadratic function in standard form is:
\[ f(x) = x^2 - 6x + 8 \]