二次方程式图像：$y=ax^2 + c$

Remember that the general form for a quadratic expression is: $$y=ax^2+bx+c$$

where $x$ is unknown (a variable), $a$ and $b$ are coefficients (numbers in front of the variable), and $c$ is a constant (a number by itself).

We also know that $a \neq 0$. We previously saw the quadratic equation when $b=0$ and $c=0$.

However, what if just $b=0$? If that's the case, we end up with this type of quadratic equation: $$y=ax^2+c$$

Let's take a look at the graph of this quadratic equation when $c=0$ and $a=1$. This graph is of the function $y=x^2$:

Now, what do you think happens when $c\neq 0$? Let's check out a few scenarios to see if we can identify a pattern.

Here is the graph for that quadratic expression, $y=x^2+5$:

The graph above is what we get when $a=1$ and $c=5$. Notice how the vertex is at $y=5$.

The graph below is what we get when $a=1$ and $c=-5$. Notice how the vertex is at $y=-5$. This is the graph of $y=x^2-5$.

So, we can say that the $y$-value of the vertex is determined by the value of $c$.

Let's leave $c=-5$, but now say that $a=-1$. What happens now? Does the value of $c$ still determine the $y$-value of the vertex??

Let's check it out! Here is a graph of $y=-x^2-5$.

Yup, the value of $c$ still determines the $y$-value of the vertex. It still opens at $-5$ because $c=-5$.

Note that the direction of the parabola (whether it opens upwards or downwards) is determined by whether or not coefficient "$a$" is positive or negative.

The coefficient $a$ in front of the $x^2$ term is positive $(\frac23)$, so the parabola will open upwards. This means the parabola will have a vertex of $(0,4)$ and open upwards.

Let's look at a graph and try to find the equation.

The correct answer here must be Choice $D$. What we know, just looking at the graph is:

• The parabola opens downwards. This means that the coefficient "$a$" must be negative.
• The parabola opens at $y=5$. This means that the quadratic equation will end in $+5$.

The only choice that fits these criteria is choice $D$: $y=-3x^2+5$.

Graphing $y=ax^2+c$

The graphs of $y=ax^2$ and $y=ax^2+c$ are almost identical. The only difference between these functions is the $c$ term. The $ax^2$ term will graph the same as if there was no $c$ term. So what does $c$ change? Let's go step by step and find the difference.

Let's begin with two functions: $$y=2x^2, y=2x^2 + 1$$ These two functions only differ by the $c$ term; here $c=1$. Lets plot $3$ points on for each function.: $x=-2, x=0,$ and $x=1$.

The function $y=2x^2$ is in blue and $y=2x^2+1$ is in red.

The red parabola is the same as the blue, but its shifted up the $y$-axis $1$ unit. This holds true for any point on each parabola.

Plotting a function with a plus $c$ term only shifts the original function up or down the $y$-axis. It is because of this property, that the $c$ term is also called the $y$-intercept. $(0,c)$ is the point that the function will cross the $y$-axis.

(Video) Graphing Quadratic Functions - 1

by mathman1024

This by mathman1024 covers graphing quadratic functions.

$\large{\text{Summary}}$

A quadratic function is a function with a degree of two. In other words, if the highest power is something squared. The general shape of a quadratic function is a parabola, or u-shape opening upward.

Once you have the basic graph you look at the components of the function and see how it gets transformed. In the video he looks at the function $f(x)=-x^2+6$. A number outside the squared term tells you whether the graph shifts up (plus) or down (minus). In this problem we have $+6$ which means we shift the graph up six units.

Now notice the second component, the minus sign in front of $x^2$. A minus sign flips the graph and makes it open downward.