Quad Test 1 Review

Answer explanation

The equation f(x) = 3 - (x + 3)^2 can be rewritten as f(x) = - (x + 3)^2 + 3, which matches the first answer choice. The negative sign indicates a reflection over the x-axis, maintaining the same vertex.

Complete the square and write in Vertex Form 
f(x) = x² + 4x + 9

• f(x) = (x+2)² + 5

• f(x) = (x - 2)² + 5

• f(x) = (x+4)² + 5

• f(x) = (x+2)² - 5

Answer explanation

To complete the square for f(x) = x² + 4x + 9, we rewrite it as f(x) = (x+2)² + 5. The vertex form is f(x) = (x+2)² + 5, confirming the correct choice is f(x) = (x+2)² + 5.

Answer explanation

To convert to vertex form, we complete the square. Starting with y = 6(x^2 + 2x) + 13, we add and subtract 1 inside the parentheses: y = 6((x+1)^2 - 1) + 13 = 6(x+1)^2 + 7. Thus, the correct answer is y=6(x+1)^2+7.

Find the vertex, focus, and directrix of the parabola: x² = 28y

• Vertex: (0, 0) Focus: (0,7) Directrix: y=-7

• Vertex: (0, 0) Focus: (7,0) Directrix: y=7

• Vertex: (0, 0) Focus: (0,-7) Directrix: x=-7

• Vertex: (0, 0) Focus: (7, 0) Directrix: x=7

Answer explanation

The equation x^2 = 28y represents a parabola that opens upwards. The vertex is at (0, 0). The focus is at (0, 7) and the directrix is y = -7, which matches the correct answer choice.

What type of parabola is described here: It has a focus of (2, -3) and a directrix of y = 7

• A vertical parabola that opens up

• A horizontal parabola that opens left

• A vertical parabola that opens down

• A horizontal parabola that opens right

Answer explanation

The focus (2, -3) is below the directrix y = 7, indicating the parabola opens downwards. Since the focus is below the directrix, it confirms that this is a vertical parabola that opens down.

Tags

CCSS.HSF-IF.C.7A

Answer explanation

To simplify \( f(x) = \frac{1}{3}(3x-9)^2 + 1 \), factor out \( 3 \) from the expression: \( (3(x-3))^2 = 9(x-3)^2 \). Thus, it becomes \( f(x) = 3(x-3)^2 + 1 \), matching the correct choice.

A quadratic function has a vertex of (2, 3) and the point (0, -9) is on the parabola. Which other point must also be on the parabola?

• (-9, 0)

• (4, -9)

• (-2, -9)

• (0, -4)

Answer explanation

The vertex (2, 3) indicates symmetry. Since (0, -9) is on the left, the corresponding point on the right is found by reflecting across the vertex, resulting in (4, -9) being on the parabola.

Tags

CCSS.HSF-IF.C.7A

• The graph has a maximum.

• The vertex of the function is at (c, d).

• The axis of symmetry is at x = c

• The focus is on the x-axis.

Answer explanation

The focus of a quadratic function in this form is not on the x-axis; it is located at (c, d + 1/(4a)), which is above the x-axis since a < 0. Thus, the statement 'The focus is on the x-axis' is NOT true.

Tags

CCSS.HSF-IF.C.7A

Answer explanation

To find the equivalent equation, expand the original: f(x) = -((1/3)x - 2)^2 = -((1/9)x^2 - (4/3)x + 4). Completing the square gives f(x) = -1/9(x - 6)^2, matching the correct choice.

How do you know if a parabola has a minimum?

• When ‘a’ is positive there is a minimum

• When ‘a’ is negative there is a minimum

• When all the numbers in the equation are negative there is a minimum

• When all the numbers in the equation are positive there is a minimum

Answer explanation

A parabola opens upwards and has a minimum point when the coefficient 'a' in the quadratic equation y = ax^2 + bx + c is positive. This means the arms of the parabola extend upwards, creating a lowest point.

Tags

CCSS.HSF-IF.C.7A