Question

1. identify the maximum or minimum value and the domain and range of the graph of the function ( y = 2(x + 2)^2 - 3 ).

a. minimum value: 3
domain: all real numbers ( geq 3 )
range: all real numbers
b. maximum value: ( -3 )
domain: all real numbers ( leq -3 )
range: all real numbers
c. maximum value: 3
domain: all real numbers
range: all real numbers ( leq 3 )
d. minimum value: ( -3 )
domain: all real numbers
range: all real numbers ( geq -3 )

1. identify the vertex and the axis of symmetry of the graph of the function ( y = 2(x + 2)^2 - 4 ).

a. vertex: ( (-2, 4) );
axis of symmetry: ( x = -2 )
b. vertex: ( (2, -4) );
axis of symmetry: ( x = 2 )
c. vertex: ( (-2, -4) );
axis of symmetry: ( x = -2 )
d. vertex: ( (2, 4) );
axis of symmetry: ( x = 2 )

what are the vertex and the axis of symmetry of the equation?

1. ( y = -2x^2 + 8x - 20 )

a. vertex: ( (-2, 12) )
axis of symmetry: ( y = -2 )
b. vertex: ( (2, -12) )
axis of symmetry: ( x = 2 )
c. vertex: ( (-2, -12) )
axis of symmetry: ( x = -2 )
d. vertex: ( (2, -12) )
axis of symmetry: ( x = -12 )

what is the expression in factored form?

1. ( x^2 + 14x + 48 )

a. ( (x + 6)(x - 8) )
b. ( (x + 8)(x - 6) )
c. ( (x - 8)(x - 6) )
d. ( (x + 6)(x + 8) )

what is the expression in factored form?

1. ( 2x^2 + 16x + 30 )

a. ( 2(x - 3)(x - 5) )
b. ( 2(x - 3)(x + 5) )
c. ( 2(x + 3)(x - 5) )
d. ( 2(x + 3)(x + 5) )

Explanation:

Question 16

Step1: Analyze the function form

The function \( y = 2(x + 2)^2 - 3 \) is in vertex form \( y=a(x - h)^2 + k \), where \( a = 2 \), \( h=-2 \), \( k = - 3 \).

Step2: Determine minimum/maximum

Since \( a=2>0 \), the parabola opens upward, so it has a minimum value at \( k=-3 \).

Step3: Determine domain

For any quadratic function, the domain is all real numbers (x can be any real number).

Step4: Determine range

Since the minimum value is -3 and the parabola opens upward, the range is all real numbers greater than or equal to -3 (\( y\geq - 3 \)).

Question 17

Step1: Recall vertex form properties

For a function in vertex form \( y = a(x - h)^2 + k \), the vertex is \( (h,k) \) and the axis of symmetry is \( x = h \).

Step2: Identify h and k

In the function \( y=2(x + 2)^2-4 \), we can rewrite \( (x + 2) \) as \( (x-(-2)) \), so \( h=-2 \) and \( k = - 4 \).

Step3: Find vertex and axis of symmetry

The vertex is \( (h,k)=(-2,-4) \) and the axis of symmetry is \( x = h=-2 \).

Question 18

Step1: Use the formula for vertex of a quadratic

For a quadratic function \( y=ax^{2}+bx + c \), the x - coordinate of the vertex is \( x=-\frac{b}{2a} \). For \( y=-2x^{2}+8x - 20 \), \( a=-2 \), \( b = 8 \).

Step2: Calculate x - coordinate of vertex

\( x=-\frac{8}{2\times(-2)}=-\frac{8}{-4} = 2 \)

Step3: Calculate y - coordinate of vertex

Substitute \( x = 2 \) into the function: \( y=-2(2)^{2}+8(2)-20=-2\times4 + 16-20=-8 + 16-20=-12 \)

Step4: Determine axis of symmetry

The axis of symmetry of a quadratic function \( y = ax^{2}+bx + c \) is \( x=-\frac{b}{2a} \), which we found to be \( x = 2 \).

Question 19

Answer:

d. minimum value: -3; domain: all real numbers; range: all real numbers ≥ -3