Use the image to answer the question.

Question

A coordinate plane's x-axis ranges from negative 10 to 2 and its y-axis ranges from negative 4 to 4, both by 1-unit increments. A concave down parabola is drawn passing through the 2nd, 3rd, and 4th quadrants.

Which response is the correct way to write the equation in vertex form for the function?

(1 point)
Responses

y=−14(x−2)2+4
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x minus 2 right parenthesis squared plus 4

y=−14(x+4)2+2
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x plus 4 right parenthesis squared plus 2

y=−14(x−4)2+2
y equals negative Start Fraction 1 over 4 End Fraction left parenthesis x minus 4 right parenthesis squared plus 2

y=−14(x+2)2+4

Answer 1

To identify the correct equation in vertex form for the concave down parabola described, we need to understand the vertex form of a quadratic function, which is written as:

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

where \((h, k)\) is the vertex of the parabola.

Since the parabola is described as concave down, the value of \(a\) must be negative.

From the information given about the coordinate plane, it appears that the vertex of the parabola is either at \((2, 4)\) or something close to that, as \(y\) is given to range from -4 to 4 and is seen at those coordinates. The parabola passes through the 2nd, 3rd, and 4th quadrants, which further suggests that the vertex might be in the 2nd quadrant.

Given the choices:

1. \(y = -\frac{1}{4}(x - 2)^2 + 4\)
2. \(y = -\frac{1}{4}(x + 4)^2 + 2\)
3. \(y = -\frac{1}{4}(x - 4)^2 + 2\)
4. \(y = -\frac{1}{4}(x + 2)^2 + 4\)

• Choice 1: This has vertex \((2, 4)\), which matches with our assumption and is concave down.

• Choice 2: Vertex at \((-4, 2)\), doesn't fit the described vertex location.

• Choice 3: Vertex at \((4, 2)\), also doesn't fit.

• Choice 4: Vertex at \((-2, 4)\), again does not match.

The correct equation is:

y = -\frac{1}{4}(x - 2)^2 + 4 (Option 1).