QQuestionOther Subjects
QuestionOther Subjects
h of x equals, 2 times, open parenthesis, x minus 4, close parenthesis, squared, minus 32
The quadratic function *h* is defined as shown. In the *xy*-plane, the graph of *y* = *h*(*x*) intersects the *x*-axis at the points (0, 0) and (1, 0), where *t* is a constant.
What is the value of *t*?
- A. 1
- B. 2
- C. 4
- D. 8
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Answer
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Step 1:: Find the quadratic equation given the x-intercepts
Since the graph of the function $h(x)$ intersects the x-axis at the points $(0, 0)$ and $(1, 0)$, we can find the quadratic equation using the x-intercepts formula: $h(x) = k(x-x_1)(x-x_2)$, where $x_1$ and $x_2$ are the x-intercepts and $k$ is a constant.
Step 2:: Plug in the given x-intercepts
Plug in the given x-intercepts, $(0, 0)$ and $(1, 0)$, into the formula: $h(x) = k(x-0)(x-1)$.
Step 3:: Simplify the equation
Simplify the equation: $h(x) = kx(x-1)$.
Step 4:: Compare the simplified equation with the given quadratic function
Compare the simplified equation with the given quadratic function $h(x) = 2(x-4)^2 - 32$.
We can see that the leading coefficient is different, so we need to adjust our equation to match the given one.
Step 5:: Find the value of $k$
Since the leading coefficient of $x^2$ is 2, we have $k = 2$.
Step 6:: Plug in the value of $k$ into the simplified equation
Plug in the value of $k$ into the simplified equation: $h(x) = 2x(x-1)$.
Step 7:: Expand the simplified equation
Expand the simplified equation: $h(x) = 2x^2 - 2x$.
Step 8:: Compare the expanded equation with the given quadratic function
Compare the expanded equation with the given quadratic function $h(x) = 2(x-4)^2 - 32$.
We can see that the linear term is different, so we need to adjust our equation to match the given one.
Step 9:: Complete the square for the simplified equation
Complete the square for the simplified equation: $h(x) = 2(x^2 - x + \frac{1}{4}) - 2(\frac{1}{4}) - 32$.
Step 10:: Simplify the adjusted equation
Simplify the adjusted equation: $h(x) = 2(x-\frac{1}{2})^2 - \frac{129}{4}$.
Step 11:: Compare the simplified adjusted equation with the given quadratic function
Compare the simplified adjusted equation with the given quadratic function $h(x) = 2(x-4)^2 - 32$.
We can see that the constant term is different, so we need to adjust our equation to match the given one.
Step 12:: Find the value of the constant term
To find the value of the constant term, we can subtract the constant term of our adjusted equation from the constant term of the given equation: $-\frac{129}{4} - (-32) = -\frac{129}{4} + 32 = \boxed{\frac{85}{4}}$.
Step 13:: Compare the simplified adjusted equation with the given quadratic function (again)
Compare the simplified adjusted equation with the given quadratic function $h(x) = 2(x-4)^2 - 32$.
We can see that the constant term is still different, so we need to adjust our equation to match the given one.
Step 14:: Find the value of $t$
Solving for $t$, we get $t = \boxed{8}$.
Final Answer
The value of $t$ is 8 (option D).
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