William Curry

2021-12-11

Determine the vertex and axis of symmetry of the following quadratics.

a)$f\left(x\right)=2(x-3)2-1$

b)$f\left(x\right)={x}^{2}+4x+3$

a)

b)

Esta Hurtado

Beginner2021-12-12Added 39 answers

Step 1

For a quadratic equation$y=a{x}^{2}+bx+c$

Axis of Symmetry is$x=\frac{-b}{2a}$

x-coordinate of vertex is$\frac{-b}{2a}$

a)$f\left(x\right)=2{(x-3)}^{2}-1$

$=2({x}^{2}-6x+9)-1$

$f\left(x\right)=2{x}^{2}-12x+17$

Axis of symmetry$\Rightarrow x=\frac{-b}{2a}$

$\Rightarrow x=-\frac{(-12)}{2\left(2\right)}=3$

$\Rightarrow x=3$

Step 2

x-coordinate of vertex$=\frac{-b}{2a}=3$

y-coordinate of vertex$=f\left(\frac{-b}{2a}\right)=f\left(3\right)$

$=2{(3-3)}^{2}-1=-1$

$\therefore$ Vertex $=(3,\text{}-1)$

b)$f\left(x\right)={x}^{2}+4x+3$

Axis of symmetry

$\Rightarrow x=\frac{-b}{2a}=\frac{-4}{2\left(1\right)}=-2$

$\Rightarrow x=-2$

x-coordinate of vertex$=\frac{-b}{2a}=-2$

Step 3

y-coordinate of vertex

$=f\left(\frac{-b}{2a}\right)$

$=f(-2)$

$={(-2)}^{2}+4(-2)+3$

$=-1$

$\therefore$ Vertex $=(-2,\text{}-1)$

For a quadratic equation

Axis of Symmetry is

x-coordinate of vertex is

a)

Axis of symmetry

Step 2

x-coordinate of vertex

y-coordinate of vertex

b)

Axis of symmetry

x-coordinate of vertex

Step 3

y-coordinate of vertex

Donald Cheek

Beginner2021-12-13Added 41 answers

Step 1

$f\left(x\right)=2(x-3)2-1$

Multiply 2 and 2 to get 4.

$4(x-3)-1$

Use the distributive property to multiply 4 by$x-3$ .

$4x-12-1$

Subtract 1 from -12 to get -13

$4x-13$

Step 2

$f\left(x\right)={x}^{2}+4x+3$

Factor the expression by grouping. First, the expression needs to be rewritten as${x}^{2}+ax+bx+3$ . To find a and b, set up a system to be solved

$a+b=4$

$ab=1\times 3=3$

Since ab is positive, a and b have the same sign. Since$a+b$ is positive, a and b are both positive. The only such pair is the system solution.

$a=1$

$b=3$

Rewrite${x}^{2}+4x+3$ as $({x}^{2}+x)+(3x+3)$

$({x}^{2}+x)+(3x+3)$

Factor out x in the first and 3 in the second group.

$x(x+1)+3(x+1)$

Factor out common term$x+1$ by using distributive property.

$(x+1)(x+3)$

Multiply 2 and 2 to get 4.

Use the distributive property to multiply 4 by

Subtract 1 from -12 to get -13

Step 2

Factor the expression by grouping. First, the expression needs to be rewritten as

Since ab is positive, a and b have the same sign. Since

Rewrite

Factor out x in the first and 3 in the second group.

Factor out common term