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

User Bhito
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1 Answer

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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
\(\left(-(7)/(2), -4.5\right)\).


User KodeTitan
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