QuestionSketch the graph of $f(x)=(x-2)^{2}+4$ using its vertex and intercepts. Find the axis of symmetry, domain, and range.

Studdy Solution

STEP 1

Assumptions1. The given function is a quadratic function, specifically a parabola. . The function is in the form $f(x) = (x - h)^ + k$ , where $(h, k)$ is the vertex of the parabola.
3. The vertex of the parabola is the point where the parabola reaches its maximum or minimum value.
4. The $y$ -intercept is the point where the graph crosses the $y$ -axis.
5. The axis of symmetry of a parabola is the vertical line through the vertex.

STEP 2

First, we identify the vertex of the parabola. The vertex is given by the point $(h, k)$ , where $h$ and $k$ are the values in the function $f(x) = (x - h)^2 + k$ .
In this case, the function is $f(x) = (x -2)^2 +4$ .
So, the vertex is $(2,4)$ .

STEP 3

Next, we find the $y$ -intercept. The $y$ -intercept is the value of $f(x)$ when $x =0$ .
So, we substitute $x =0$ into the function to find the $y$ -intercept.
$f(0) = (0 -2)^2 +$

STEP 4

Calculate the $y$ -intercept.
$f(0) = (0 -2)^2 +4 =4 +4 =8$ So, the $y$ -intercept is $(0,8)$ .

STEP 5

Now, we find the equation of the axis of symmetry. The axis of symmetry of a parabola is the vertical line $x = h$ , where $h$ is the $x$ -coordinate of the vertex.
In this case, the vertex is $(2,4)$ , so the axis of symmetry is $x =2$ .

STEP 6

Next, we identify the domain and range of the function.The domain of a function is the set of all possible $x$ -values. For a quadratic function, the domain is all real numbers, or $(-\infty, \infty)$ .
The range of a function is the set of all possible $y$ -values. For a parabola that opens upwards (as in this case), the range is $[k, \infty)$ , where $k$ is the $y$ -coordinate of the vertex.In this case, the vertex is $(2,4)$ , so the range is $[4, \infty)$ .

STEP 7

Finally, we can sketch the graph of the function using the vertex, $y$ -intercept, and axis of symmetry.1. Plot the vertex $(2,4)$ .
2. Plot the $y$ -intercept $(0,)$ .
3. Draw the axis of symmetry $x =2$ .
4. Since the parabola opens upwards, draw the parabola starting at the vertex and passing through the $y$ -intercept, symmetric about the axis of symmetry.

Was this helpful?

QuestionSketch the graph of $f(x)=(x-2)^{2}+4$ using its vertex and intercepts. Find the axis of symmetry, domain, and range.

Studdy Solution

STEP 1

STEP 2

First, we identify the vertex of the parabola. The vertex is given by the point $(h, k)$ , where $h$ and $k$ are the values in the function $f(x) = (x - h)^2 + k$ .
In this case, the function is $f(x) = (x -2)^2 +4$ .
So, the vertex is $(2,4)$ .

STEP 3

Next, we find the $y$ -intercept. The $y$ -intercept is the value of $f(x)$ when $x =0$ .
So, we substitute $x =0$ into the function to find the $y$ -intercept.
$f(0) = (0 -2)^2 +$

STEP 4

Calculate the $y$ -intercept.
$f(0) = (0 -2)^2 +4 =4 +4 =8$ So, the $y$ -intercept is $(0,8)$ .

STEP 5

Now, we find the equation of the axis of symmetry. The axis of symmetry of a parabola is the vertical line $x = h$ , where $h$ is the $x$ -coordinate of the vertex.
In this case, the vertex is $(2,4)$ , so the axis of symmetry is $x =2$ .

STEP 6

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSketch the graph of f(x)=(x−2)2+4f(x)=(x-2)^{2}+4f(x)=(x−2)2+4 using its vertex and intercepts. Find the axis of symmetry, domain, and range.

Studdy Solution

STEP 1

STEP 2

STEP 3

Next, we find the yyy-intercept. The yyy-intercept is the value of f(x)f(x)f(x) when x=0x =0x=0.So, we substitute x=0x =0x=0 into the function to find the yyy-intercept.f(0)=(0−2)2+f(0) = (0 -2)^2 +f(0)=(0−2)2+

STEP 4

Calculate the yyy-intercept.f(0)=(0−2)2+4=4+4=8f(0) = (0 -2)^2 +4 =4 +4 =8f(0)=(0−2)2+4=4+4=8So, the yyy-intercept is (0,8)(0,8)(0,8).

STEP 5

Now, we find the equation of the axis of symmetry. The axis of symmetry of a parabola is the vertical line x=hx = hx=h, where hhh is the xxx-coordinate of the vertex.In this case, the vertex is (2,4)(2,4)(2,4), so the axis of symmetry is x=2x =2x=2.

STEP 6

STEP 7

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

QuestionSketch the graph of $f(x)=(x-2)^{2}+4$ using its vertex and intercepts. Find the axis of symmetry, domain, and range.

Next, we find the $y$ -intercept. The $y$ -intercept is the value of $f(x)$ when $x =0$ .
So, we substitute $x =0$ into the function to find the $y$ -intercept.
$f(0) = (0 -2)^2 +$

Calculate the $y$ -intercept.
$f(0) = (0 -2)^2 +4 =4 +4 =8$ So, the $y$ -intercept is $(0,8)$ .

Now, we find the equation of the axis of symmetry. The axis of symmetry of a parabola is the vertical line $x = h$ , where $h$ is the $x$ -coordinate of the vertex.
In this case, the vertex is $(2,4)$ , so the axis of symmetry is $x =2$ .