Question

1-6
do you understand?

1. essential question how does the equation of a quadratic function in vertex form highlight key features of the functions graph?
2. error analysis given the function $g(x)=(x+3)^2$, martin says the graph should be translated right 3 units from the parent graph $f(x)=x^2$. explain his error.
3. vocabulary what shape does a quadratic function have when graphed?
4. communicate precisely how are the graphs of $f(x)=x^2$ and $g(x)=-(x+2)^2-4$ related?

do you know how?
describe the transformation of the parent function $f(x)=x^2$.

1. $g(x)=(x+5)^2+2$
2. $h(x)=(x-2)^2-7$

write the equation of each parabola in vertex form.

1. vertex: $(-3, 7)$; point: $(-2, -5)$
2. vertex: $(1, 3)$; point: $(2, 5)$
3. vertex: $(-4, 6)$; point: $(-2, -2)$
4. vertex: $(7, 4)$; point: $(5, 16)$

Explanation:

Step1: Solve Q1 (Vertex Form Feature)

Vertex form $f(x)=a(x-h)^2+k$ directly gives vertex $(h,k)$ and direction of opening from $a$.

Step2: Solve Q2 (Martin's Error)

For $g(x)=(x+3)^2=(x-(-3))^2$, horizontal shift is left 3 units, not right. Martin misread the sign inside the parentheses: shifts opposite the sign inside $(x-h)$.

Step3: Solve Q3 (Quadratic Graph Shape)

Quadratic functions graph to a parabola (U-shaped curve).

Step4: Solve Q4 (Graph Relationship)

$g(x)=-(x+2)^2-4$: reflect $f(x)=x^2$ over x-axis, shift left 2, down 4.

Step5: Solve Q5 (Transform $g(x)=(x+1)^2+2$)

Parent $f(x)=x^2$: shift left 1, up 2 units.

Step6: Solve Q6 (Transform $h(x)=(x-2)^2-7$)

Parent $f(x)=x^2$: shift right 2, down 7 units.

Step7: Solve Q7 (Vertex Form Equation)

Vertex form: $y=a(x-h)^2+k$, vertex $(-3,7)$ so $y=a(x+3)^2+7$. Substitute $(-2,-5)$:
$-5=a(-2+3)^2+7 \implies -5=a(1)+7 \implies a=-12$. Equation: $y=-12(x+3)^2+7$

Step8: Solve Q8 (Vertex Form Equation)

Vertex $(1,3)$: $y=a(x-1)^2+3$. Substitute $(2,5)$:
$5=a(2-1)^2+3 \implies 5=a+3 \implies a=2$. Equation: $y=2(x-1)^2+3$

Step9: Solve Q9 (Vertex Form Equation)

Vertex $(-4,6)$: $y=a(x+4)^2+6$. Substitute $(-2,-2)$:
$-2=a(-2+4)^2+6 \implies -2=4a+6 \implies 4a=-8 \implies a=-2$. Equation: $y=-2(x+4)^2+6$

Step10: Solve Q10 (Vertex Form Equation)

Vertex $(7,4)$: $y=a(x-7)^2+4$. Substitute $(5,16)$:
$16=a(5-7)^2+4 \implies 16=4a+4 \implies 4a=12 \implies a=3$. Equation: $y=3(x-7)^2+4$

Answer:

1. The vertex form $f(x)=a(x-h)^2+k$ directly shows the vertex $(h,k)$ and the direction/width of the parabola from the value of $a$.
2. Martin made an error in the direction of the horizontal shift: $g(x)=(x+3)^2=(x-(-3))^2$ is a left 3-unit shift from $f(x)=x^2$, not right. Horizontal shifts are opposite the sign inside the parentheses.
3. A quadratic function graphs as a parabola (a symmetric U-shaped or upside-down U-shaped curve).
4. The graph of $g(x)=-(x+2)^2-4$ is the graph of $f(x)=x^2$ reflected over the x-axis, shifted 2 units to the left, and shifted 4 units down.
5. $g(x)=(x+1)^2+2$ is the parent function $f(x)=x^2$ shifted 1 unit left and 2 units up.
6. $h(x)=(x-2)^2-7$ is the parent function $f(x)=x^2$ shifted 2 units right and 7 units down.
7. $y=-12(x+3)^2+7$
8. $y=2(x-1)^2+3$
9. $y=-2(x+4)^2+6$
10. $y=3(x-7)^2+4$