Question

Find the​ y-intercept, the axis of​ symmetry, and the vertex of the graph of the function.
​f(x)equals2 x squared plus 6 x plus 10
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Part 1
The​ y-intercept is
  
left parenthesis 0 comma 10 right parenthesis. ​(Type an ordered​ pair.)
Part 2
The axis of symmetry is
  
enter your response here. ​(Simplify your answer. Type an​ equation.)

Answer 1

To find the y-intercept, axis of symmetry, and vertex of the given quadratic function \( f(x) = 2x^2 + 6x + 10 \), we can follow these steps:

### Part 1: Y-Intercept
The y-intercept is found by evaluating the function at \( x = 0 \):

\[
f(0) = 2(0)^2 + 6(0) + 10 = 10
\]

So, the y-intercept is \( (0, 10) \).

### Part 2: Axis of Symmetry
The axis of symmetry for a quadratic function in the form \( f(x) = ax^2 + bx + c \) can be found using the formula:

\[
x = -\frac{b}{2a}
\]

Here, \( a = 2 \) and \( b = 6 \):

\[
x = -\frac{6}{2 \cdot 2} = -\frac{6}{4} = -\frac{3}{2}
\]

Thus, the axis of symmetry is:

\[
x = -\frac{3}{2}
\]

### Part 3: Vertex
The vertex can be found using the axis of symmetry. To find the y-coordinate of the vertex, substitute \( x = -\frac{3}{2} \) back into the function:

\[
f\left(-\frac{3}{2}\right) = 2\left(-\frac{3}{2}\right)^2 + 6\left(-\frac{3}{2}\right) + 10
\]
\[
= 2\left(\frac{9}{4}\right) - 9 + 10
\]
\[
= \frac{18}{4} - 9 + 10
\]
\[
= \frac{18}{4} - \frac{36}{4} + \frac{40}{4}
\]
\[
= \frac{18 - 36 + 40}{4} = \frac{22}{4} = \frac{11}{2}
\]

So, the vertex is at:

\[
\left(-\frac{3}{2}, \frac{11}{2}\right)
\]

### Summary:
- The y-intercept is \( (0, 10) \).
- The axis of symmetry is \( x = -\frac{3}{2} \).
- The vertex is indicated by the ordered pair \( \left(-\frac{3}{2}, \frac{11}{2}\right) \).

If you need to input only the y-intercept and axis of symmetry:
- \( (0, 10) \)
- \( x = -\frac{3}{2} \)