identify the vertex, axis of symmetry, and min/max value of each.
18) $f(x) = 2x^2 - 3$
19) $f(x) = -x^2 - 10x$
20) $f(x) = 3x^2 + 24x + 42$
21) $f(x) = -(x - 3)^2$
22) $f(x) = (x - 2)^2 + 1$

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Accepted Answer

# Explanation: ## Step1: Analyze $ f(x) = 2x^2 - 3 $ The vertex form of a parabola is $ f(x)=a(x - h)^2 + k $, where $(h,k)$ is the vertex. For $ f(x)=2x^2 - 3 $, we can rewrite it as $ f(x)=2(x - 0)^2 - 3 $. So $ h = 0 $, $ k=-3 $, vertex is $(0, -3)$. The axis of symmetry is $ x = h = 0 $. Since $ a = 2>0 $, the parabola opens up, so it has a minimum value at $ k=-3 $. ## Step2: Analyze $ f(x)=-x^2 - 10x $ Complete the square: $ f(x)=-(x^2 + 10x)=-(x^2 + 10x + 25 - 25)=-(x + 5)^2 + 25 $. So vertex is $(-5, 25)$, axis of symmetry $ x=-5 $. Since $ a=-1<0 $, it opens down, maximum value is $ 25 $. ## Step3: Analyze $ f(x)=3x^2 + 24x + 42 $ Factor out 3: $ f(x)=3(x^2 + 8x)+42 $. Complete the square inside the parentheses: $ x^2 + 8x=(x + 4)^2 - 16 $. So $ f(x)=3((x + 4)^2 - 16)+42=3(x + 4)^2 - 48 + 42=3(x + 4)^2 - 6 $. Vertex $(-4, -6)$, axis of symmetry $ x=-4 $. $ a = 3>0 $, minimum value $-6$. ## Step4: Analyze $ f(x)=-(x - 3)^2 - \dots $ (assuming the missing part doesn't affect form analysis). Vertex form is $ f(x)=a(x - h)^2 + k $, so vertex $(3, k)$ (depending on the missing term), axis of symmetry $ x = 3 $. Since $ a=-1<0 $, it opens down, maximum at $ k $. ## Step5: Analyze $ f(x)=(x - 2)^2 + 1 $ In vertex form $ f(x)=1(x - 2)^2 + 1 $, so vertex $(2, 1)$, axis of symmetry $ x = 2 $. $ a = 1>0 $, minimum value $ 1 $. # Answer: 18) Vertex: $(0, -3)$, Axis of Symmetry: $ x = 0 $, Minimum Value: $-3$ 19) Vertex: $(-5, 25)$, Axis of Symmetry: $ x=-5 $, Maximum Value: $ 25 $ 20) Vertex: $(-4, -6)$, Axis of Symmetry: $ x=-4 $, Minimum Value: $-6$ 21) (Assuming form $ f(x)=-(x - 3)^2 + k $) Vertex: $(3, k)$, Axis of Symmetry: $ x = 3 $, Maximum Value: $ k $ (depends on missing term) 22) Vertex: $(2, 1)$, Axis of Symmetry: $ x = 2 $, Minimum Value: $ 1 $

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Explanation:

Step1: Analyze \( f(x) = 2x^2 - 3 \)

Step2: Analyze \( f(x)=-x^2 - 10x \)

Step3: Analyze \( f(x)=3x^2 + 24x + 42 \)

Step4: Analyze \( f(x)=-(x - 3)^2 - \dots \) (assuming the missing part doesn't affect form analysis). Vertex form is \( f(x)=a(x - h)^2 + k \), so vertex \((3, k)\) (depending on the missing term), axis of symmetry \( x = 3 \). Since \( a=-1<0 \), it opens down, maximum at \( k \).

Step5: Analyze \( f(x)=(x - 2)^2 + 1 \)

Answer: