Question

writing quadratic functions in factored form homework
work the following problems on a separate sheet of paper
writing in intercept form write a quadratic function in intercept form for
the parabola shown.
17.
19.
writing in intercept form write a quadratic function in intercept form
whose graph has the given x-intercepts and passes through the given point.

1. x-intercepts: 2, 5
2. x-intercepts: -3, 0
3. x-intercepts: -1, 4

point: (4, -2)
point: (2, 10)
point: (2, 4)

1. * multiple choice the x-intercepts of a parabola are 4 and 7 and another

point on the parabola is (2, -20). which point is also on the parabola?
ⓐ (1, 21)
ⓑ (8, -4)
ⓒ (5, -40)
ⓓ (5, 4)
error analysis describe and correct the error in writing a quadratic function
whose graph has the given x-intercepts or vertex and passes through the given
point.

1. x-intercepts: 4, -3; point: (5, -5)
2. vertex: (2, 3); point: (1, 5)

$y = a(x - 5)(x + 5)$
$y = a(x - 2)(x - 3)$
$-3 = a(4 - 5)(4 + 5)$
$5 = a(1 - 2)(1 - 3)$
$-3 = -9a$
$5 = 2a$
$\frac{1}{3} = a$, so $y = \frac{1}{3}(x - 5)(x + 5)$
$\frac{5}{2} = a$, so $y = \frac{5}{2}(x - 2)(x - 3)$

1. write a quadratic function in vertex form
2. write a quadratic equation in standard

whose graph has vertex at (3, 5) and passes
for the parabola that passes through t
through the point (7, -3).
points (-2, -13), (2, 3), and (4, 5).
writing in standard form write the quadratic function in standard form.

1. $y = -3(x - 2)(x - 4)$
2. $f(x) = 5(x + 3)^2 - 4$

factored or intercept form:

Explanation:

Let's solve problem 14: Write a quadratic function in vertex form whose graph has vertex at \((3, 5)\) and passes through the point \((7, -3)\).

Step 1: Recall Vertex Form

The vertex form of a quadratic function is \( y = a(x - h)^2 + k \), where \((h, k)\) is the vertex of the parabola. Here, the vertex is \((3, 5)\), so \( h = 3 \) and \( k = 5 \). Substituting these into the vertex form, we get:
\( y = a(x - 3)^2 + 5 \)

Step 2: Substitute the Given Point

We know the parabola passes through the point \((7, -3)\). This means when \( x = 7 \), \( y = -3 \). Substitute these values into the equation from Step 1 to solve for \( a \):
\( -3 = a(7 - 3)^2 + 5 \)

Step 3: Solve for \( a \)

Simplify the equation:
First, calculate \( (7 - 3)^2 \):
\( (7 - 3)^2 = 4^2 = 16 \)
So the equation becomes:
\( -3 = 16a + 5 \)
Subtract 5 from both sides:
\( -3 - 5 = 16a \)
\( -8 = 16a \)
Divide both sides by 16:
\( a = \frac{-8}{16} = -\frac{1}{2} \)

Step 4: Write the Final Equation

Now that we have the value of \( a \), substitute it back into the vertex form equation from Step 1:
\( y = -\frac{1}{2}(x - 3)^2 + 5 \)

Answer:

\( y = -\frac{1}{2}(x - 3)^2 + 5 \)