QuestionGraph $f(x) = x^2 - 4x$ . Determine if it opens up or down, and find the vertex, axis of symmetry, $y$ -intercept, and $x$ -intercepts.

Studdy Solution

STEP 1

Assumptions1. The function given is a quadratic function, $f(x) = x^ -4x$ . We need to find the vertex, axis of symmetry, $y$ -intercept, and $x$ -intercepts of the function3. The quadratic function is in the form $f(x) = ax^ + bx + c$ , where $a$ , $b$ , and $c$ are constants

STEP 2

First, let's identify the coefficients $a$ , $b$ , and $c$ in the given quadratic function. $f(x) = x^2 -4x$ Here, $a =1$ , $b = -4$ , and $c =0$ .

STEP 3

Next, we will determine whether the graph of the function opens up or down. This depends on the sign of $a$ . If $a >0$ , the graph opens up. If $a <0$ , the graph opens down.
Since $a =1$ which is greater than0, the graph of the function opens up.

STEP 4

Now, let's find the vertex of the function. The vertex of a quadratic function $f(x) = ax^2 + bx + c$ is given by the point $(-\frac{b}{2a}, f(-\frac{b}{2a}))$ .

STEP 5

Plug in the values for $a$ and $b$ to find the $x$ -coordinate of the vertex.
$x = -\frac{b}{2a} = -\frac{-4}{2 \cdot1} =2$

STEP 6

Now, substitute $x =2$ into the function to find the $y$ -coordinate of the vertex.
$y = f(2) = (2)^2 -4(2) =4 -8 = -4$ So, the vertex of the function is $(2, -4)$ .

STEP 7

The axis of symmetry of a quadratic function is the vertical line $x = -\frac{b}{2a}$ .
So, the axis of symmetry of the function is $x =2$ .

STEP 8

The $y$ -intercept of a function is the point where the graph of the function intersects the $y$ -axis. This occurs when $x =0$ .
So, substitute $x =0$ into the function to find the $y$ -intercept.
$y = f(0) = (0)^2 -4(0) =0$ So, the $y$ -intercept of the function is $(0,0)$ .

STEP 9

The $x$ -intercepts of a function are the points where the graph of the function intersects the $x$ -axis. This occurs when $f(x) =$ .
So, set the function equal to zero and solve for $x$ to find the $x$ -intercepts.
$= x^2 -4x$

STEP 10

Factor the equation to solve for $x$ .
$0 = x(x -4)$ Setting each factor equal to zero gives the solutions $x =0$ and $x =4$ .
So, the $x$ -intercepts of the function are $(0,0)$ and $(4,0)$ .
The graph of the function $f(x) = x^2 -4x$ opens up, its vertex is at $(2, -4)$ , its axis of symmetry is $x =2$ , its $y$ -intercept is $(0,0)$ , and its $x$ -intercepts are $(0,0)$ and $(4,0)$ .

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