Notes
| \(\text{ Parabolas }\) | |||
| \(\underline{\text{Vertex Form}}\) | \(\underline{\text{Standard Form}}\) | \(\underline{\text{Intercepts Form}}\) | |
|---|---|---|---|
| \(\text{Equation}\) | |||
| \(\text{Vertex}\) | |||
| \(\text{Axis of Symmetry}\) | |||
| \(\text{Opening Direction}\) | \( \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\) |
||
Problems
State the vertex, axis of symmetry and opening direction.
\(\textbf{1)}\) \(f(x)=3(x-4)^2+2\) Hint:
This is vertex form \(\rightarrow f(x)=a(x-h)^2+k\)
Vertex \(\rightarrow (h,k)\)
Axis of Symmetry \(\rightarrow x=h\)
Direction of Opening \(\rightarrow \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\)
The vertex is \( (4,2) \) The axis of symmetry is \( x=4 \) The opening direction is up
This is vertex form \(\rightarrow f(x)=a(x-h)^2+k\)
Vertex \(\rightarrow (h,k)\)
Axis of Symmetry \(\rightarrow x=h\)
Direction of Opening \(\rightarrow \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\)
\(\textbf{2)}\) \(f(x)=-2(x+3)^2-1\) The vertex is \( (-3,-1) \) The axis of symmetry is \( x=-3 \) The opening direction is down
\(\textbf{3)}\) \(f(x)=(x+5)^2-3\) The vertex is \( (-5,-3) \) The axis of symmetry is \( x=-5 \) The opening direction is up
\(\textbf{4)}\) \(f(x)=4(x-4)^2\) The vertex is \( (4,0) \) The axis of symmetry is \( x=4 \) The opening direction is up
\(\textbf{5)}\) \(f(x)=-x^2+3\) The vertex is \( (0,3) \) The axis of symmetry is \( x=0 \) The opening direction is down
\(\textbf{6)}\) \(f(x)=2x^2-4x+5\) Hint:
This is standard form \(f(x)=ax^2+bx+c, \,\,\,\)
Vertex \(=\rightarrow \left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)\)
Axis of Symmetry \(\rightarrow x=\frac{-b}{2a}\)
Direction of Opening \(\rightarrow \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\)
The vertex is \( (1,3) \) The axis of symmetry is \( x=1 \) The opening direction is up
This is standard form \(f(x)=ax^2+bx+c, \,\,\,\)
Vertex \(=\rightarrow \left(\frac{-b}{2a},f\left(\frac{-b}{2a}\right)\right)\)
Axis of Symmetry \(\rightarrow x=\frac{-b}{2a}\)
Direction of Opening \(\rightarrow \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\)
\(\textbf{7)}\) \(f(x)=x^2-6x+3\) The vertex is \( (3,-6) \) The axis of symmetry is \( x=3 \) The opening direction is up
\(\textbf{8)}\) \(f(x)=-x^2+4x+5\) The vertex is \( (2,9) \) The axis of symmetry is \( x=2 \) The opening direction is down
\(\textbf{9)}\) \(f(x)=4x^2+8x-2\) The vertex is \( (-1,-6) \) The axis of symmetry is \( x=-1 \) The opening direction is up
\(\textbf{10)}\) \(f(x)=-2x^2+8x-3\) The vertex is \( (2,5) \) The axis of symmetry is \( x=2 \) The opening direction is down
\(\textbf{11)}\) \(f(x)=2(x-5)(x-1)\) Hint:
This is intercept form \(f(x)=a(x-p)(x-q), \,\,\,\)
Vertex \(\rightarrow \left(\frac{p+q}{2},f\left(\frac{p+q}{2}\right)\right)\)
Axis of Symmetry \(\rightarrow x=\frac{p+q}{2}\)
Direction of Opening \(\rightarrow \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\)
The vertex is \( (3,-8) \) The axis of symmetry is \( x=3 \) The opening direction is up
This is intercept form \(f(x)=a(x-p)(x-q), \,\,\,\)
Vertex \(\rightarrow \left(\frac{p+q}{2},f\left(\frac{p+q}{2}\right)\right)\)
Axis of Symmetry \(\rightarrow x=\frac{p+q}{2}\)
Direction of Opening \(\rightarrow \text{ opens up if }a\gt0, \text{ opens down if } a\lt0\)
\(\textbf{12)}\) \(f(x)=-3(x+5)(x-3)\) The vertex is \( (-1,48) \) The axis of symmetry is \( x=-1 \) The opening direction is down
\(\textbf{13)}\) \(f(x)=5(x)(x-4)\) The vertex is \( (2,-20) \) The axis of symmetry is \( x=2 \) The opening direction is up
\(\textbf{14)}\) \(f(x)=-(x+7)(x+5)\) The vertex is \( (-6,1) \) The axis of symmetry is \( x=-6 \) The opening direction is down
\(\textbf{15)}\) \(f(x)=4(x)(x-2)\) The vertex is \( (1,-4) \) The axis of symmetry is \( x=1 \) The opening direction is up
See Related Pages\(\)
\(\bullet\text{ All Conic Section Notes}\)
\(\,\,\,\,\,\,\,\,\)
\(\bullet\text{ Equation of a Circle}\)
\(\,\,\,\,\,\,\,\,(x-h)^2+(y-k)^2=r^2…\)
\(\bullet\text{ Parabolas}\)
\(\,\,\,\,\,\,\,\,y=a(x-h)^2+k…\)
\(\bullet\text{ Axis of Symmetry}\)
\(\,\,\,\,\,\,\,\,x=-\frac{b}{2a}…\)
\(\bullet\text{ Ellipses}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}+\frac{(y-k)^2}{b^2}=1…\)
\(\bullet\text{ Area of Ellipses}\)
\(\,\,\,\,\,\,\,\,\text{Area}=\pi a b…\)
\(\bullet\text{ Hyperbolas}\)
\(\,\,\,\,\,\,\,\,\frac{(x-h)^2}{a^2}-\frac{(y-k)^2}{b^2}=1…\)
\(\bullet\text{ Conic Sections- Completing the Square}\)
\(\,\,\,\,\,\,\,\,x^2+8x+y^2−6y=11 \Rightarrow (x+4)^2+(y−3)^2=36…\)
\(\bullet\text{ Conic Sections- Parametric Equations}\)
\(\,\,\,\,\,\,\,\,x=h+r \cos{t}\)
\(\,\,\,\,\,\,\,\,y=k+r \sin{t}…\)
\(\bullet\text{ Degenerate Conics}\)
\(\,\,\,\,\,\,\,\,x^2−y^2=0…\)
\(\bullet\text{ Andymath Homepage}\)
