Question

identify the x-intercepts, the y-intercept, and the vertex of the graph of the function y=2x^2 +14x +20

Answer 1

The y-intercept is at (0, 20).

The vertex of the quadratic function
`\(y = 2x^2 + 14x + 20\)` is
`\(\left(-(7)/(2), -4.5\right)\).`

To identify the x-intercepts, y-intercept, and vertex of the quadratic function
`\(y = 2x^2 + 14x + 20\)`, let's break down each component:

1. **X-intercepts:**

X-intercepts occur where the graph intersects the x-axis, i.e., where y = 0. To find them, set y to 0 and solve for x:

`\[ 2x^2 + 14x + 20 = 0 \]`

You can use the quadratic formula to solve for
`\(x\)`, which is
`\((-b \pm √(b^2 - 4ac))/(2a)\)` in this case.

2. **Y-intercept:**

The y-intercept is where the graph intersects the y-axis, and for any function, it occurs when x = 0. Substituting x = 0 into the function gives the y-intercept:

`\[ y = 2(0)^2 + 14(0) + 20 = 20 \]`

3. **Vertex:**

The vertex of a quadratic function in the form
`\(y = ax^2 + bx + c\)` is given by the coordinates
`\((- (b)/(2a), f(-(b)/(2a)))\). For \(y = 2x^2 + 14x + 20\)`, the x-coordinate of the vertex is
`\(-(14)/(4) = -(7)/(2)\)`, and substituting this into the function gives the y-coordinate:

`\[ y = 2\left(-(7)/(2)\right)^2 + 14\left(-(7)/(2)\right) + 20 \]`

simplify the expression step by step:

`\[ y = 2\left(-(7)/(2)\right)^2 + 14\left(-(7)/(2)\right) + 20 \]`

1. Square the term inside the parentheses:

`\[ = 2 * (49)/(4) + 14\left(-(7)/(2)\right) + 20 \]`

2. Multiply 2 by \(\frac{49}{4}\):

`\[ = (98)/(4) + 14\left(-(7)/(2)\right) + 20 \]`

3. Simplify the fraction:

`\[ = 24.5 - 49 + 20 \]`

4. Combine like terms:

`\[ = -4.5 \]`

Therefore, the y-coordinate of the vertex is -4.5.

So, the simplified expression is:

`\[ y = -4.5 \]`

The vertex of the quadratic function
`\(y = 2x^2 + 14x + 20\)` is
`\(\left(-(7)/(2), -4.5\right)\).`