(3.4) Factored Form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$
Graph the following functions on Desmos above.
$f\left(x\right)=\left(x-1\right)\left(x-5\right)$
$g\left(x\right)=\left(x+2\right)\left(x+4\right)$
$h\left(x\right)=\left(x+1\right)\left(x-4\right)$
$x$-intercepts
Write down the coordinates of the $x$-intercepts. Separate your answers with a comma.
$f\left(x\right)$
$g\left(x\right)$
$h\left(x\right)$
$y$-intercept
To find the $y$-intercept of a function, we substitute $x=$
Write down the coordinates of the $y$-intercept for each function.
$f\left(x\right)$
$g\left(x\right)$
$h\left(x\right)$
[Practice] Write down the coordinates of the $x$-intercepts of the following functions. Separate your answers with a comma.
$f\left(x\right)=\left(x+2\right)\left(x+4\right)$
$f\left(x\right)=\left(x-4\right)\left(x-3\right)$
$f\left(x\right)=\left(x+4\right)\left(x+6\right)$
$f\left(x\right)=\left(x+5\right)\left(x-4\right)$
[Practice] Write down the coordinates of the $y$-intercept of the following functions.
$f\left(x\right)=\left(x-2\right)\left(x-8\right)$
$f\left(x\right)=\left(x+4\right)\left(x+1\right)$
$f\left(x\right)=\left(x-8\right)\left(x-6\right)$
$f\left(x\right)=\left(x-3\right)\left(x+5\right)$
Axis of symmetry
Draw the axis of symmetry for each function on the grid above and write the equations below.
$f\left(x\right)$
$g\left(x\right)$
The axis of symmetry passes through the midpoint between the two $\fbox{ _ - i _ _ _ _ _ _ _ _ _ }$
Find the equation of the axis of symmetry of $h\left(x\right)=\left(x+1\right)\left(x-4\right)$. Check this on your graph.
Vertex
Since the vertex lies on the axis of symmetry,
$f\left(x\right)=\left(x-1\right)\left(x-5\right)$
axis of symmetry: $x=3$
vertex: $\left(3,\square\right)$
$g\left(x\right)=\left(x+2\right)\left(x+4\right)$
axis of symmetry: $x=-3$
vertex: $\left(-3,\square\right)$
$h\left(x\right)=\left(x+1\right)\left(x-4\right)$
axis of symmetry: $x=1.5$
vertex: $\left(1.5,\square\right)$
vertex:
vertex:
vertex:
[Practice] Write down the equation of the axis of symmetry for the following functions.
$f\left(x\right)=\left(x-8\right)\left(x+4\right)$
$f\left(x\right)=\left(x-4\right)\left(x+2\right)$
$f\left(x\right)=\left(x-6\right)\left(x+3\right)$
$f\left(x\right)=\left(x-9\right)\left(x+9\right)$
[Practice] Write down the coordinates of the vertex for the following functions.
$f\left(x\right)=\left(x+5\right)\left(x-5\right)$
$f\left(x\right)=\left(x+3\right)\left(x+6\right)$
Leading Coefficient, $a$
Graph the following functions on the grid above.
$f\left(x\right)=\left(x-1\right)\left(x-5\right)$
$g\left(x\right)=2\left(x-1\right)\left(x-5\right)$
$h\left(x\right)=-\frac{1}{3}\left(x-1\right)\left(x-5\right)$
The coordinates of the 𝑥-intercepts for all three graphs are:
The equation of the axis of symmetry for all three graphs is:
For a function written in the form $f\left(x\right)=a\left(x-p\right)\left(x-q\right)$,
If $a$ is positive, the parabola is concave $\fbox{u _}$.
If $a$ is negative, the parabola is concave $\fbox{d _ _ _}$.
[Practice] For the function $f\left(x\right)=-2\left(x+3\right)\left(x-1\right)$, find:
the coordinates of the $x$-intercepts. Separate your answers with a comma.
the coordinates of the $y$-intercept.
the equation of the axis of symmetry.
the coordinates of the vertex.
the concavity of the parabola. Write "concave up" or "concave down".
[Practice] For the function $f\left(x\right)=3x^2+9x-30$, find:
it in the form $a\left(x-p\right)\left(x-q\right)\qquad f\left(x\right)=$
the coordinates of the $x$-intercepts. Separate your answers with a comma.
the coordinates of the $y$-intercept.
the equation of the axis of symmetry.
the coordinates of the vertex.
the concavity of the parabola. Write "concave up" or "concave down".
[Practice] For the function $f\left(x\right)=\frac{1}{4}x^2+3x+8$, find:
it in the form $a\left(x-p\right)\left(x-q\right)\qquad f\left(x\right)=$
the coordinates of the $x$-intercepts. Separate your answers with a comma.
the coordinates of the $y$-intercept.
the equation of the axis of symmetry.
the coordinates of the vertex.
the concavity of the parabola. Write "concave up" or "concave down".