1. **Write the equation $f(x) = x^2 + 2x - 4$ in vertex form.** The vertex form of a quadratic is $$f(x) = a(x-h)^2 + k$$ where $(h,k)$ is the vertex. 2. Complete the square for $x^2 + 2x - 4$: $$f(x) = x^2 + 2x - 4 = (x^2 + 2x + 1) - 1 - 4 = (x + 1)^2 - 5$$ 3. So the vertex form is $$f(x) = (x + 1)^2 - 5$$ --- 1. **Write the equation $f(x) = -2x^2 - 6x$ in factored form.** 2. Factor out the greatest common factor (GCF): $$f(x) = -2x^2 - 6x = -2x(x + 3)$$ 3. So the factored form is $$f(x) = -2x(x + 3)$$ --- 1. **Identify the transformation for the graph of $f(x) = x^2$ shifted to vertex $(-1, -3)$.** 2. The transformation is $$f(x+1) - 3$$ which shifts the graph left by 1 and down by 3. 3. Among the options, the correct one is $$f(x+1) - 3$$ (not listed exactly but matches the description). --- 1. **Determine if $f(x) = 2x^2 - 4x + 8$ has a maximum at $(1,6)$.** 2. The coefficient of $x^2$ is positive (2), so the parabola opens upwards and has a minimum, not a maximum. 3. Therefore, the statement is **false**. --- 1. **Explain the transformations for $-2f(x+1) - 3$.** 2. $f(x)$ is the parent function $x^2$. 3. $f(x+1)$ shifts left by 1. 4. Multiplying by $-2$ reflects vertically and stretches by factor 2. 5. Subtracting 3 shifts down by 3. 6. So the correct description is: vertical reflection and stretch by a factor of 2, left 1, down 3. --- 1. **Find the roots of $h(x) = 2x^2 - 8x$.** 2. Factor: $$h(x) = 2x(x - 4)$$ 3. Set each factor to zero: $$2x = 0 \Rightarrow x=0$$ $$x - 4 = 0 \Rightarrow x=4$$ 4. Roots are $(0,0)$ and $(4,0)$. --- 1. **Find the vertex of $f(x) = -x^2 + 5x + 5$.** 2. Use vertex formula $x = -\frac{b}{2a}$: $$a = -1, b = 5$$ $$x = -\frac{5}{2(-1)} = \frac{5}{2} = 2.5$$ 3. Find $f(2.5)$: $$f(2.5) = -(2.5)^2 + 5(2.5) + 5 = -6.25 + 12.5 + 5 = 11.25$$ 4. Vertex is at $(2.5, 11.25)$. --- Final answers: - Vertex form: $f(x) = (x + 1)^2 - 5$ - Factored form: $f(x) = -2x(x + 3)$ - Transformation: $f(x+1) - 3$ - Maximum point statement: false - Transformations for $-2f(x+1) - 3$: vertical reflection and stretch by 2, left 1, down 3 - Roots of $h(x)$: $(0,0)$ and $(4,0)$ - Vertex of $f(x) = -x^2 + 5x + 5$: $(2.5, 11.25)$