Solved on Sep 19, 2023

Find the axis of symmetry, vertex, and y-intercept of the quadratic function $f(x) = -x^2 - 8x + 5$ .

STEP 1

Assumptions1. The function is given as $f(x)=-x^{}-8 x+5$ . The function is a quadratic function, which has the form $f(x)=ax^{}+bx+c$ where a, b, and c are constants.
3. The axis of symmetry of a quadratic function is given by the formula $x=-\frac{b}{a}$
4. The vertex of a quadratic function is given by the point $(h, k)$ where $h=-\frac{b}{a}$ and $k=f(h)$ 5. The y-intercept of a quadratic function is the point where the graph of the function intersects the y-axis. This happens when $x=0$ .

STEP 2

First, we need to find the axis of symmetry. We can do this by using the formula $x=-\frac{b}{2a}$ .

STEP 3

Now, plug in the given values for a and b to calculate the axis of symmetry.
$x=-\frac{-8}{2(-1)}$

STEP 4

Calculate the axis of symmetry.
$x=-\frac{-8}{-2} =4$

STEP 5

Now that we have the axis of symmetry, we can find the y-coordinate of the vertex by plugging the x-coordinate of the vertex (which is the axis of symmetry) into the function.
$k=f(4)=-4^{2}-8(4)+5$

STEP 6

Calculate the y-coordinate of the vertex.
$k=-16-32+5=-43$

STEP 7

Now that we have the coordinates of the vertex, we can write them as a point (h, k).
$Vertex = (4, -43)$

STEP 8

Now, we need to find the y-intercept. We can do this by plugging x=0 into the function.
$f(0)=-0^{2}-8(0)+5$

STEP 9

Calculate the y-intercept.
$f()=5$ The y-intercept is the point where the graph of the function intersects the y-axis, so it is the point (, f()).
$y-intercept = (,5)$ The axis of symmetry is $x=4$ , the vertex is $(4, -43)$ , and the y-intercept is $(,5)$ .

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Find the axis of symmetry, vertex, and y-intercept of the quadratic function $f(x) = -x^2 - 8x + 5$ .

STEP 1

STEP 2

First, we need to find the axis of symmetry. We can do this by using the formula $x=-\frac{b}{2a}$ .

STEP 3

Now, plug in the given values for a and b to calculate the axis of symmetry.
$x=-\frac{-8}{2(-1)}$

STEP 4

Calculate the axis of symmetry.
$x=-\frac{-8}{-2} =4$

STEP 5

Now that we have the axis of symmetry, we can find the y-coordinate of the vertex by plugging the x-coordinate of the vertex (which is the axis of symmetry) into the function.
$k=f(4)=-4^{2}-8(4)+5$

STEP 6

Calculate the y-coordinate of the vertex.
$k=-16-32+5=-43$

STEP 7

Now that we have the coordinates of the vertex, we can write them as a point (h, k).
$Vertex = (4, -43)$

STEP 8

Now, we need to find the y-intercept. We can do this by plugging x=0 into the function.
$f(0)=-0^{2}-8(0)+5$

STEP 9

Calculate the y-intercept.
$f()=5$ The y-intercept is the point where the graph of the function intersects the y-axis, so it is the point (, f()).
$y-intercept = (,5)$ The axis of symmetry is $x=4$ , the vertex is $(4, -43)$ , and the y-intercept is $(,5)$ .

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Find the axis of symmetry, vertex, and y-intercept of the quadratic function f(x)=−x2−8x+5f(x) = -x^2 - 8x + 5f(x)=−x2−8x+5.

STEP 1

STEP 2

First, we need to find the axis of symmetry. We can do this by using the formula x=−b2ax=-\frac{b}{2a}x=−2ab​.

STEP 3

Now, plug in the given values for a and b to calculate the axis of symmetry.x=−−82(−1)x=-\frac{-8}{2(-1)}x=−2(−1)−8​

STEP 4

Calculate the axis of symmetry.x=−−8−2=4x=-\frac{-8}{-2} =4x=−−2−8​=4

STEP 5

Now that we have the axis of symmetry, we can find the y-coordinate of the vertex by plugging the x-coordinate of the vertex (which is the axis of symmetry) into the function.k=f(4)=−42−8(4)+5k=f(4)=-4^{2}-8(4)+5k=f(4)=−42−8(4)+5

STEP 6

Calculate the y-coordinate of the vertex.k=−16−32+5=−43k=-16-32+5=-43k=−16−32+5=−43

STEP 7

Now that we have the coordinates of the vertex, we can write them as a point (h, k).Vertex=(4,−43)Vertex = (4, -43)Vertex=(4,−43)

STEP 8

Now, we need to find the y-intercept. We can do this by plugging x=0 into the function.f(0)=−02−8(0)+5f(0)=-0^{2}-8(0)+5f(0)=−02−8(0)+5

STEP 9

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Find the axis of symmetry, vertex, and y-intercept of the quadratic function $f(x) = -x^2 - 8x + 5$ .

First, we need to find the axis of symmetry. We can do this by using the formula $x=-\frac{b}{2a}$ .

Now, plug in the given values for a and b to calculate the axis of symmetry.
$x=-\frac{-8}{2(-1)}$

Calculate the axis of symmetry.
$x=-\frac{-8}{-2} =4$

Now that we have the axis of symmetry, we can find the y-coordinate of the vertex by plugging the x-coordinate of the vertex (which is the axis of symmetry) into the function.
$k=f(4)=-4^{2}-8(4)+5$

Calculate the y-coordinate of the vertex.
$k=-16-32+5=-43$

Now that we have the coordinates of the vertex, we can write them as a point (h, k).
$Vertex = (4, -43)$

Now, we need to find the y-intercept. We can do this by plugging x=0 into the function.
$f(0)=-0^{2}-8(0)+5$