Math / Algebra

QuestionFill in the blanks Enter intercepts as ordered pairs, aka points. Then Graph the parabola given in standard form. You ony need to graph the vertex and one other point.
Standard Form: $f(x)=-x^{2}-4 x$
1. Does the parabola open up or down? O Up Down
2. Vertex $(x, y)=$ $\square$
3. $y$ -intercept $(x, y))=$ $\square$
4. Equation of the Axis of Symmetry: $\square$ Clear mill Draw: $\square$

Studdy Solution

STEP 1

What is this asking? We're finding the vertex and intercepts of a parabola, then graphing it with just the vertex and one more point! Watch out! Don't forget: the parabola opens down because of the negative sign in front of $x^2$ !

STEP 2

1. Determine the direction of the parabola
2. Find the vertex
3. Find the $y$ -intercept
4. Determine the equation of the axis of symmetry
5. Graph the parabola

STEP 3

Alright, let's look at the equation $f(x) = -x^2 - 4x$ .
The key here is the negative sign in front of $x^2$ .
This tells us the parabola opens down!

STEP 4

To find the vertex, we use the formula for the vertex of a parabola in standard form $ax^2 + bx + c$ .
The $x$ -coordinate of the vertex is given by:
$x = -\frac{b}{2a}$ Here, $a = -1$ and $b = -4$ .
Plug these into the formula:
$x = -\frac{-4}{2 \cdot (-1)} = -\frac{4}{-2} = 2$

STEP 5

Now, let's find the $y$ -coordinate by substituting $x = 2$ back into the function:
$f(2) = -(2)^2 - 4 \cdot 2 = -4 - 8 = -12$ So, the vertex is at $(2, -12)$ .

STEP 6

The $y$ -intercept is where the graph crosses the $y$ -axis, which happens when $x = 0$ .
Substitute $x = 0$ into the function:
$f(0) = -(0)^2 - 4 \cdot 0 = 0$ So, the $y$ -intercept is at $(0, 0)$ .

STEP 7

The axis of symmetry is a vertical line that passes through the vertex.
Its equation is $x =$ the $x$ -coordinate of the vertex.
So, the axis of symmetry is:
$x = 2$

STEP 8

To graph, plot the vertex $(2, -12)$ and the $y$ -intercept $(0, 0)$ .
Draw the parabola opening downwards, making sure it's symmetric about the line $x = 2$ .

STEP 9

1. The parabola opens down.
2. Vertex $(x, y) = (2, -12)$
3. $y$ -intercept $(x, y) = (0, 0)$
4. Equation of the Axis of Symmetry: $x = 2$

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Studdy Solution

STEP 1

What is this asking? We're finding the vertex and intercepts of a parabola, then graphing it with just the vertex and one more point! Watch out! Don't forget: the parabola opens down because of the negative sign in front of $x^2$ !

STEP 2

1. Determine the direction of the parabola
2. Find the vertex
3. Find the $y$ -intercept
4. Determine the equation of the axis of symmetry
5. Graph the parabola

STEP 3

Alright, let's look at the equation $f(x) = -x^2 - 4x$ .
The key here is the negative sign in front of $x^2$ .
This tells us the parabola opens down!

STEP 4

To find the vertex, we use the formula for the vertex of a parabola in standard form $ax^2 + bx + c$ .
The $x$ -coordinate of the vertex is given by:
$x = -\frac{b}{2a}$ Here, $a = -1$ and $b = -4$ .
Plug these into the formula:
$x = -\frac{-4}{2 \cdot (-1)} = -\frac{4}{-2} = 2$

STEP 5

Now, let's find the $y$ -coordinate by substituting $x = 2$ back into the function:
$f(2) = -(2)^2 - 4 \cdot 2 = -4 - 8 = -12$ So, the vertex is at $(2, -12)$ .

STEP 6

The $y$ -intercept is where the graph crosses the $y$ -axis, which happens when $x = 0$ .
Substitute $x = 0$ into the function:
$f(0) = -(0)^2 - 4 \cdot 0 = 0$ So, the $y$ -intercept is at $(0, 0)$ .

STEP 7

The axis of symmetry is a vertical line that passes through the vertex.
Its equation is $x =$ the $x$ -coordinate of the vertex.
So, the axis of symmetry is:
$x = 2$

STEP 8

To graph, plot the vertex $(2, -12)$ and the $y$ -intercept $(0, 0)$ .
Draw the parabola opening downwards, making sure it's symmetric about the line $x = 2$ .

STEP 9

1. The parabola opens down.
2. Vertex $(x, y) = (2, -12)$
3. $y$ -intercept $(x, y) = (0, 0)$
4. Equation of the Axis of Symmetry: $x = 2$

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Studdy Solution

STEP 1

What is this asking? We're finding the vertex and intercepts of a parabola, then graphing it with just the vertex and one more point! Watch out! Don't forget: the parabola opens down because of the negative sign in front of x2x^2x2!

STEP 2

1. Determine the direction of the parabola2. Find the vertex3. Find the yyy-intercept4. Determine the equation of the axis of symmetry5. Graph the parabola

STEP 3

Alright, let's look at the equation f(x)=−x2−4xf(x) = -x^2 - 4xf(x)=−x2−4x.The key here is the **negative sign** in front of x2x^2x2.This tells us the parabola opens **down**!

STEP 4

STEP 5

Now, let's find the yyy-coordinate by substituting x=2x = 2x=2 back into the function:f(2)=−(2)2−4⋅2=−4−8=−12f(2) = -(2)^2 - 4 \cdot 2 = -4 - 8 = -12f(2)=−(2)2−4⋅2=−4−8=−12So, the vertex is at (2,−12)(2, -12)(2,−12).

STEP 6

The yyy-intercept is where the graph crosses the yyy-axis, which happens when x=0x = 0x=0.Substitute x=0x = 0x=0 into the function:f(0)=−(0)2−4⋅0=0f(0) = -(0)^2 - 4 \cdot 0 = 0f(0)=−(0)2−4⋅0=0So, the yyy-intercept is at (0,0)(0, 0)(0,0).

STEP 7

The axis of symmetry is a vertical line that passes through the vertex.Its equation is x=x =x= the xxx-coordinate of the vertex.So, the axis of symmetry is:x=2x = 2x=2

STEP 8

To graph, plot the vertex (2,−12)(2, -12)(2,−12) and the yyy-intercept (0,0)(0, 0)(0,0).Draw the parabola opening downwards, making sure it's symmetric about the line x=2x = 2x=2.

STEP 9

1. The parabola opens **down**.2. Vertex (x,y)=(2,−12)(x, y) = (2, -12)(x,y)=(2,−12)3. yyy-intercept (x,y)=(0,0)(x, y) = (0, 0)(x,y)=(0,0)4. Equation of the Axis of Symmetry: x=2x = 2x=2

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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.

What is this asking? We're finding the vertex and intercepts of a parabola, then graphing it with just the vertex and one more point! Watch out! Don't forget: the parabola opens down because of the negative sign in front of $x^2$ !

1. Determine the direction of the parabola
2. Find the vertex
3. Find the $y$ -intercept
4. Determine the equation of the axis of symmetry
5. Graph the parabola

Alright, let's look at the equation $f(x) = -x^2 - 4x$ .
The key here is the negative sign in front of $x^2$ .
This tells us the parabola opens down!

Now, let's find the $y$ -coordinate by substituting $x = 2$ back into the function:
$f(2) = -(2)^2 - 4 \cdot 2 = -4 - 8 = -12$ So, the vertex is at $(2, -12)$ .

The $y$ -intercept is where the graph crosses the $y$ -axis, which happens when $x = 0$ .
Substitute $x = 0$ into the function:
$f(0) = -(0)^2 - 4 \cdot 0 = 0$ So, the $y$ -intercept is at $(0, 0)$ .

The axis of symmetry is a vertical line that passes through the vertex.
Its equation is $x =$ the $x$ -coordinate of the vertex.
So, the axis of symmetry is:
$x = 2$

To graph, plot the vertex $(2, -12)$ and the $y$ -intercept $(0, 0)$ .
Draw the parabola opening downwards, making sure it's symmetric about the line $x = 2$ .

1. The parabola opens down.
2. Vertex $(x, y) = (2, -12)$
3. $y$ -intercept $(x, y) = (0, 0)$
4. Equation of the Axis of Symmetry: $x = 2$