The graph of the function f left parenthesis x right parenthesis equals ax squared plus bx plus c has its vertex at left parenthesis 0 comma 1 right parenthesis and passes through the point left parenthesis 1 comma 9 right parenthesis .  Find​ a, b, and c.

We are given the quadratic function \( f(x) = ax^2 + bx + c \) and the following information:

- The vertex is at \( (0, 1) \), meaning \( f(0) = 1 \).
- The graph passes through the point \( (1, 9) \), meaning \( f(1) = 9 \).

### Step 1: Use the vertex information
Since the vertex is \( (0, 1) \), we can immediately determine \( c \) because at \( x = 0 \), \( f(0) = 1 \). So, using \( f(0) = ax^2 + bx + c \):

\[
f(0) = a(0)^2 + b(0) + c = c = 1
\]

Thus, \( c = 1 \).

### Step 2: Use the point \( (1, 9) \)
Next, we use the fact that the point \( (1, 9) \) lies on the curve, so \( f(1) = 9 \). Substituting \( x = 1 \) and \( f(1) = 9 \) into the equation:

\[
f(1) = a(1)^2 + b(1) + 1 = 9
\]
\[
a + b + 1 = 9
\]
\[
a + b = 8 \quad \text{(Equation 1)}
\]

### Step 3: Use the vertex form condition
For a parabola with vertex \( (h, k) \), the equation is given by:

\[
f(x) = a(x - h)^2 + k
\]

In this case, the vertex is \( (0, 1) \), so:

\[
f(x) = a(x - 0)^2 + 1 = ax^2 + 1
\]

Thus, from the vertex form, we know \( b = 0 \).

### Step 4: Solve for \( a \)
Now that we know \( b = 0 \), we substitute it into Equation 1:

\[
a + 0 = 8 \implies a = 8
\]

### Final Answer:
Thus, the values of \( a \), \( b \), and \( c \) are:
\[
a = 8, \quad b = 0, \quad c = 1
\]

The function is \( f(x) = 8x^2 + 1 \).

Would you like further clarification, or should I dive deeper into any part of this?

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Here are 5 related questions for practice:
1. What is the axis of symmetry of the function \( f(x) = 8x^2 + 1 \)?
2. How do you find the vertex of a quadratic function in standard form?
3. If the function passed through \( (2, 33) \) instead of \( (1, 9) \), how would that affect the equation?
4. Can a quadratic function have more than one vertex?
5. What is the domain and range of the function \( f(x) = 8x^2 + 1 \)?

**Tip:** The vertex form of a quadratic function is useful for finding the vertex directly from the equation.

The graph of the function f(x) = ax^2 + bx + c has its vertex at (0, 1) and passes through the point (1, 9). Find a, b, and c.

Solve the quadratic function f(x) = ax^2 + bx + c by using the vertex at (0, 1) and the point (1, 9). This detailed step-by-step solution involves key algebraic concepts, including vertex form and standard form of a quadratic function. Suitable for students in Grades 9-12.

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